The Reds controlled the game against their rivals from across Stanley Park, who were content to sit back in numbers under Sam Allardyce and look for a smash-and-grab result.
Mohamed Salah broke their resistance shortly before half-time with his 19th goal of the season in all competitions, a goal which appeared to put Liverpool on course for a comfortable victory.
But Sadio Mané spurned a sublime chance to make it 2-0 before half-time, dragging a shot well wide, and Liverpool could not translate their dominance into goalscoring chances in the second half.
And despite Everton's lack of attacking threat, Wayne Rooney drew the Toffees level with a 77th-minute penalty after Dejan Lovren's clumsy push on Dominic Calvert-Lewin tempted referee Craig Pawson to point to the spot.
And a stunned Liverpool side could not produce a response even despite the introductions of Philippe Coutinho and Danny Ings, having to settle for one point instead of all three.
Oh Mané, Mané...
For 77 minutes Liverpool were in complete and utter control, perhaps even more so than Klopp's first derby in April 2016 when his side enjoyed 67 per-cent possession, had 37 shots and scored four goals.
In fact, while Liverpool did not pepper the visitors' goal quite so often here - they saw almost all of the ball (79 per-cent possession) and still had 23 shots. Everton had three.
Simply, they had enough spells of pressure that they should have produced more chances and more goals. Just three shots on target was an incredibly poor return.
By the time Rooney drew Everton level with a bolt from the blue, the game should really have been over - and that it wasn't, Liverpool can only point the finger of blame at themselves.
Mané had scored in his previous two derbies, a memorable 95th minute winner at Goodison Park last December and the opener in a 3-1 win at Anfield back in April, and should have been the scourge of the Blues' Christmas again this year.
From a throw in, the winger nipped in front of Ashley Williams to reach Dominic Solanke's flick on and move the ball into the space beyond the defender.
He burst through on goal as Liverpool launched a four-on-one attack, Cuco Martina the last Everton defender and Solanke, Alex Oxlade-Chamberlain and Salah all in the box to his right.
But with the latter two in particular in swathes of space, Mané instead took the brash decision to take the shot on himself - rather than square for a tap-in.
The Senegalese international got the execution of his shot horribly wrong, dragging his left-footed effort four yards wide of the far post.
Instead of killing the game, Liverpool went into the break with the game very much in the balance. Klopp let Mané know his feelings from the touch-line.
Had Mané doubled Liverpool's advantage, Everton might have been forced to commit forward and open up the game. At least, the Toffees would have needed a mighty second-half display to get back into it from 2-0 down.
Instead Everton remained in the game, knowing always that they were one set-piece, one quick counter-attack or one defensive mistake away from a treasured equaliser. It was an avoidable Lovren mistake from which their eventual equaliser came.
In truth, aside from the Mané chance and a tricky far-post header which Salah could just guide wide from a James Milner cross, Liverpool just did not create enough clear chances from the amount of possession they enjoyed.
Coutinho forced a good save from a free-kick while Mané tried to make amends with a spectacular overhead kick from the edge of the box that Pickford saved well and Joe Gomez headed over at a free-kick. From open play there was very little to seriously trouble Everton.
Even Salah's goal was not really a chance - more a product of the Egyptian's individual brilliance, the same which has seen him score a variety of stunning goals and terrorise opposition defences throughout the campaign.
It was for that which they will regret coming away from the game. But Mané's decision not to square to his team-mates rather than go it alone ultimately proved most costly - and one he will be desperate to rectify in the coming games.
The rotation debate
The conditions on Merseyside resembled a snow globe and so perhaps it was fitting that Jürgen Klopp oversaw such a dramatic shake-up of his starting eleven.
Six changes, with Philippe Coutinho and Roberto Firmino - nine goals and three assists between them in Liverpool's last two games - both controversially dropped to the bench.
The logic was clear. The pivotal Brazilian pair have played plenty of minutes this season and Dominic Solanke offered greater physicality and aerial presence against Everton's back-line, particularly with the deliveries of left-back Andrew Robertson an option in Liverpool's attacking arsenal.
Yet even Salah's goal, coming against the run of play in a dour opening period, could not distract from the fact that Liverpool lacked creativity from midfield and rarely threatened to open up Allardyce's well-drilled charges in the first half.
It didn't improve much in the second half, Liverpool seeing so much possession and yet yielding little end product.
Liverpool's dominance can perhaps be use as a justification for Klopp's rotation, that they still were the better team, but it is certainly fair to suggest that they might have put it well beyond Everton had the 'Fab Four' of Coutinho, Firmino, Mané and Salah all started.
This was not the time to break the quartet up. They had a relatively routine evening in mid-week with the 7-0 thrashing of Spartak Moscow - in which they all scored, they were scoring for fun and combining effortlessly.
There is no doubt the absences of some of them on the team-sheet will have been a psychological boost for the Everton team, and their absences told.
Yet there is also the nagging resentment that Liverpool could, and should, have won the game despite being without some key players.
Klopp will be frustrated by the decision-making of both Mané - not to pass - and Lovren - to place his hands on the back of Calvert-Lewin and offer the striker the opportunity to go to ground.
It was a soft penalty but one which Lovren should never have invited with such stupidity and lack of foresight.
Had the two been smarter in either instance, Liverpool come away with a fully-deserved victory and leapfrog Chelsea into third. Instead they remain two points behind the Blues, only one above Arsenal - who also dropped points this weekend - and just two above Tottenham Hotspur.
And Klopp's rotation will be what many centre in on as to why Liverpool missed this opportunity to not only claim a huge win but move further ahead of some of their top-four rivals.
Not starting Coutinho and Firmino, as well as Emre Can, was an enormous risk - and one which consequently can be seen to have cost them as much as Mané and Lovren's irrational thinking.
With their best attackers on the pitch Liverpool surely would have punished Everton's guarded display before they could strike back late on.
Klopp's substitutions were also questionable. Salah, who had the better of Martina down the right flank, was a surprising withdrawal with 25 minutes to play - while Liverpool cried out for Coutinho long before he actually came on, after Rooney had netted his penalty.
Liverpool's rotation policy had worked to perfection before this, and perhaps was always likely to come unstuck. It will not distract Klopp from utilising the depth of his squad in upcoming games, considering the burn-out they suffered from overworking players last season, and rightly so - but here perhaps his rotation was naïve.
Excellent Gomez arguably the stand-out
As performances in your first Merseyside derby go, Gomez's must be among the most impressive - especially for a defender.
The 20-year-old's long-term future is seen to be at centre-back but his performance at right-back here was straight out of the top drawer.
This was arguably his finest ever performance for the Reds, his passing crisp, his tackling and interceptions well-timed (bar one second-half foul on Martina) and his running and movement intelligent.
He constantly offered himself as an option in attack, overlapping Salah, and was dependable and solid in everything he did defensively. Composed and mature, Gomez looked the best of any of Liverpool's defenders - and perhaps only Salah was as good as him of any Liverpool player to start.
He was sluggish after his initial return from a lengthy injury in January and appeared too cumbersome, but having since been on an individual training programme appears far more refined. His form has improved tenfold.
Gomez might have even scored second half, guiding a header on to the roof of the net from a Salah free-kick just past the hour with Pickford well beaten.
The England U21 captain has regularly swapped duties with Trent Alexander-Arnold this season, though this was a clear example of just what has convinced Klopp to make Gomez his favourite.
Nathaniel Clyne who?
The Henderson dilemma
"He is such an important player for us. I don’t get why I have to say that."
The words of Klopp on captain Jordan Henderson in his pre-match press conference earlier in the week.
On this evidence, he may have to repeat them. That line of questioning was in reaction to Klopp having dropped his skipper for their crucial Champions League group stage clash with Spartak Moscow on Wednesday night.
As expected, the England international returned to the team here - but his performance was lacking in influence. He alone completed more passes than the entire Everton team in the first half, but few of them more than a simple side-ways pass. He rarely ever looked like driving his team forward.
In fairness, that is not always his role as the deepest midfielder - but with James Milner alongside him, Henderson was designated the more creative of the pair. That never showed, though.
Liverpool's midfield - as it has been before with Milner and Henderson paired together - was undoubtedly less effective and missed the drive and power of Can. In his place, Henderson offered too little.
Running and an excellent work-rate is a constant with Henderson, but his passing was too often - aside from some sweeping long balls that stretched play to find the Reds' wide men - monotonous and repetitive. On several occasions a poorly-chosen choice of pass halted Liverpool's momentum, with Henderson often the perpetrator.
He comes in for criticism, much of it over the top and unwarranted, but this was a inefficient performance from the man with the armband.
Henderson's afternoon was aptly summarised by his last-minute shot, which he shanked high into the Kop to a chorus of frustrated groans.
While it is clear that Klopp still has faith in Henderson, he must work to repay such belief before the entire Liverpool fanbase begin to turn against him. A growing majority already have.
Reds must react despite their bubble being burst
It is one of those results that feels more like a defeat than a draw. Taken from the reaction of the jubilant visiting Everton fans it could have been just that.
But while the result is disappointing given Liverpool's dominance over their neighbours, it is only a minor setback. A bubble-bursting result after confidence had surged with recent wins, more so than a significant blow.
Yet they bounced back from a similarly frustrating result at home to Chelsea, poor defending late on also cancelling out a Salah equaliser that evening, and should do so again.
They are 10 games unbeaten and can continue that run and return to winning ways against Alan Pardew's West Bromwich Albion, who fell to defeat to a struggling Swansea City on Saturday, when they host the West Midlands outfit in mid-week.
The Baggies, just above the relegation zone, are the kind of opposition that Liverpool have relished playing this term - having swept aside lesser lights far more comfortably this season.
It is important to remember that while a disappointing draw against Everton, smarter decisions from certain individuals would mean a full three points to toast and another reaction of praise towards Klopp's rotation. While the performance was not perfect, it was enough to win - and should have been.
Coutinho and Firmino will come back into the line-up in mid-week, though Salah and Mané could well drop out, and Liverpool will hope the Brazilian duo can get them back to their free-scoring ways.