Summer signing Salah scored a first-half brace 10 minutes apart to level and then break Robbie Fowler's record - which had stood since December 1993 - of eight strikes in his first 12 games.
Philippe Coutinho delivered a decisive third on 68 minutes, firing low into the bottom corner from 12 yards after Fraser Forster failed to deal with Roberto Firmino's drive to complete a dominant display from the hosts.
But it was Salah who took all the plaudits for his excellent double, which makes him the Premier League's top goalscorer and takes his tally in all competitions to a phenomenal 14 goals in 18 games.
Southampton - managed by former Liverpool defender and first-team coach Mauricio Pellegrino - failed to showcase any attacking threat throughout a meek performance.
Liverpool - who have now won three successive league games for the first time in 2017 - leapfrog Arsenal into fifth place due to their superior goal difference - and sit just a point off fourth, while Southampton drop down to 14th.
Reds just too good for the Saints
Liverpool were the first to threaten after a low-key start, Salah turning Sadio Mané's cross wide on the volley before Georginio Wijnaldum and then Coutinho drew diving saves from Forster.
But Southampton's resistance - having kept four successive clean sheets against Liverpool last season and not conceded to them in 459 minutes across the previous five encounters and this one - was broken on 31 minutes.
Firmino dispossessed Dušan Tadić on the edge of the Saints' box before Wijnaldum's pass to Salah to his right appeared to take the winger out of his stride. But the Egyptian set himself to cut in and bend a sublime left-footed strike beyond Forster and into the far corner.
Simon Mignolet had to be alert to punch clear Cédric Soares' cross in towards Sofiane Boufal to protect the Reds' advantage but Salah doubled his - and Liverpool's - lead on 41 minutes.
Coutinho's defence-splitting pass gave Salah, who darted behind Maya Yoshida and Virgil van Dijk into open space, the opportunity to cleverly flick beyond the on-rushing Forster and make it 2-0.
An excellent block from ex-Saints defender Dejan Lovren prevented Boufal from creating more of a dangerous opening early in the second-half as Southampton looked to respond.
Liverpool should have put the game out of sight on the hour mark but after a four-on-one counter attack, Alberto Moreno could not finish from Salah's through ball.
Their control continued and Coutinho fired just wide before Ryan Bertrand turned Moreno's low cross away from goal with Firmino lurking.
The third goal finally came on 68 minutes when Forster could only push Firmino's drive out into the middle of the box with his chest and Coutinho - who now has 10 goals and four assists in his last 15 Liverpool outings - ran on to the loose ball to place into the bottom corner.
Stellar Salah stars again
The 'Chelsea reject' tag bandied about by some when Liverpool spent a then club-record £39.7 million to sign Salah from AS Roma in June could hardly look more foolish now.
His brace here took him top of the Premier League goalscoring charts and above the likes of Harry Kane, Romelu Lukaku and Sergio Agüero. This, for a winger, remember. And a winger only six months into his time on Merseyside and just three since his competitive debut.
Now the club's Player of the Month winner for August, September and October (shared with Lovren), Salah also has as many goals as the club's highest scorer last term - Coutinho - in six months less.
Producing the most consistent output within a glittering attack, it is difficult to do justice to how well Salah has done to become so integral to Liverpool so quickly into his Anfield career.
What the 25-year-old adds not just in goals and assists but in pace, movement and energy is worth double, if not triple, the fee that Liverpool paid for him and perhaps the scary thing is that he has not been all that prolific this season.
One Liverpool fan at the Kop end held aloft an Egypt flag as Salah approached to take a second-half corner, reminiscent of when Reds fans' appreciation for Luis Suárez was shown by the number of Uruguay flags on the Kop several years ago.
And it appears Salah, like Suárez, will be the one to spearhead Liverpool's hopes this season as they chart another top-four finish - the title surely out of reach due to Manchester City's imperiousness - and a run into the Champions League's latter stages.
Few expected him to enjoy such a blistering start but if he keeps this form up, he'll reach the 30-goal tally that only Suárez has (twice) reached for Liverpool in the last nine seasons and if he can do that, Jürgen Klopp's side stand a chance of a successful season.
He's certainly becoming a fans' favourite. That showed when Salah was rightly greeted with a rapturous standing ovation on his way off on 80 minutes, his name ringing around all four corners of Anfield. It most definitely won't be the last time that happens.
Reds' defence fares fine in Matip's absence
Liverpool's inarguable weakness, their defence, has now conceded just once in six league games at Anfield this season - and today too, they were without Joël Matip and with Joe Gomez on the bench.
The Cameroonian international was sidelined by an adductor injury which led to Lovren's first start since a disastrous display away at Tottenham Hotspur, in which he was substituted after only half-an-hour.
Lovren - as to be expected - was subject to a poor reception and several unrepeatable chants from the visiting contingent after the way he left Southampton to join Liverpool in the summer of 2014, but his performance was not lacking in confidence bar a few wayward headers and he was faultless with several key blocks, tackles and interceptions.
Alongside him Ragnar Klavan, likewise capable of error-prone performances defended, well throughout and formed a back-four which denied Southampton a single shot on target - just as on their previous league visit to Anfield in May.
Admittedly, their back-line was rarely tested here but when called upon, they performed well to snuff out the threat of the likes of Shane Long, Boufal and Tadić.
Moreno and Alexander-Arnold also performed well in either full-back position, offering welcome balance in attack and defence - the former even wasting a fine chance.
This was another encouraging performance from their entire defensive unit and they have now kept seven clean sheets in their last eight home league games, conceding only once in the sole exception to that streak - a 1-1 draw with Burnley.
If they can begin to maintain some defensive consistency away from home, and exert more of the control and professionalism of this performance, then the magnificence of their attack will see to it that they are there or thereabouts where they want to be come May.
On the opposing side, van Dijk - a defender whom Liverpool spent all summer chasing and were willing to pay as much as £75 million for - was not at his best in what many will hope isn't his only run-out at Anfield this year.
His decision-making for Liverpool's second goal was particularly questionable, rushing forward to leave a gaping hole by Maya Yoshida which Salah duly ran into to score.
He struggled - as many a defender would - up against the verve and vivacity of Liverpool's quartet of Mané, Firmino, Salah and Coutinho - who started a league game together for the first time - but that is not to say that he wouldn't add an enormous amount to the Merseyside outfit's options at the back were he to sign in January or next summer.
The Dutchman did execute a few well-timed interceptions and blocks, particularly to deny Firmino a goal late on from Alex Oxlade-Chamberlain's attempt to square.
Pellegrino under pressure?
Southampton came into this game with as many points as they had at the same stage last season, but it is increasingly difficult to see them replicating their eighth-placed finish of 2016-17.
Their coming festive fixtures pit them against all of last season's other top six teams with daunting trips to City, Chelsea, Spurs and Manchester United all to face.
At this rate, lacking any sort of clearly-defined attacking plan - their possession-based approach yielding very few chances and even any attacks of note - they will be fortunate not to be beaten in all four of those games.
Their performance began promisingly but quickly faded away and the only time they had Mignolet truly scrambling was from Bertand's first-half free-kick, which he bent just wide of the post. Charlie Austin's late header was another half-chance.
Southampton have finished in the top 10 of the Premier League in all of the last four seasons, but the boos after their defeat to Burnley at St Mary's before the international break represented the discontent among their fan base.
Having sacked Claude Puel - who reached a first major final in 14 years - for Southampton's lack of attacking threat in the summer, Pellegrino has yet to prove he can add the entertainment and excitement that he promised upon his arrival.
They scored just 41 goals in 38 games under Puel last season but little has changed yet this term, with just nine goals in 12 top-flight games and three of those coming from penalties. They have only scored two goals in a game once in all competitions.
Pellegrino - who has had mixed spells of success as a manager but did excellently at Deportivo Alavés in his previous job before moving to Southampton - is in trouble if this form, but more so the manner of his side's performances, continues.