In an almost unstoppable exhibition of attacking football, the Reds were 3-0 up by half-time thanks to goals from Sadio Mane, Philippe Coutinho and Emre Can.
Roberto Firmino added a fourth, before Mane's second of the game made it five after 60 minutes.
Daryl Janmaat pulled one back for Watford, but could not stop Liverpool delivering another huge statement of intent in the wake of Chelsea's 5-0 win over Everton on Saturday.
Second-half substitute Georginio Wijnaldum rounded the scoring off, tapping in from close-range late on to seal Klopp's biggest ever home win and put Liverpool at the summit of the English top-flight for at least the next two weeks.
Two changes for Liverpool, just the one for Watford
Jürgen Klopp made two changes to the team that beat Crystal Palace last weekend, with centre-back Dejan Lovren ruled out with a virus.
Lucas Leiva stepped in for the Croatian for just his second league start of the season, while James Milner - himself out ill last weekend - reprised his left-back role over Alberto Moreno.
Emre Can kept his spot in midfield over Georginio Wijnaldum, while Daniel Sturridge and Divock Origi remaind second choice to Roberto Firmino up top.
For the visitors, Walter Mazzarri made just the one change with defender Sebastian Prodl replaced by Daryl Janmaat, the Austrian still suffering from an abductor injury sustained in the Hornets' win over Hull a week ago.
Reds go close a number of times before first goal
Watford arrived on Merseyside having kept three successive clean sheets and retained the 3-5-2 formation that had helped them climb up to seventh in the table after 10 league games.
But like any other team to have visited Anfield this season, they struggled to shackle all of Liverpool's varying forward threats. Early on Adam Lallana drove into space down the left and crossed in to Firmino, but he drilled his into the ground and Heurelho Gomes needed only catch it.
Nordin Amrabat fired a tame shot towards Loris Karius up the other end as Watford occasionally looked to press their opponents high up the pitch, but much of the early stages were played in Watford's half.
Liverpool went into the game knowing they could go top of the league with just a win, and played with the conviction of a side determined to do so.
Having twice scored from corner-kicks in their win at Palace, Lucas Leiva squandered a golden chance from another to break the deadlock on 15 minutes.
Philippe Coutinho's delivery dropped perfectly for the Brazilian at the back post, but though he had time to pick his place Lucas' unfamiliarity in front of goal showed, Gomes denying him well to further extend his 191-game wait for a goal.
From there, Liverpool regularly breached Watford's back-line in their search for the game's opening goal, Coutinho crashing the crossbar from close-range although he was offside.
Minutes later, the No.10 - so often the driving force for Liverpool this term - came close again when his fine first touch from a Mane pass took him beyond Janmaat and into space in the area, but his low effort struck the side-netting of the near post.
Mane, Coutinho and Can all on the scoresheet before half-time
The opening goal seemed only a matter of time, with Klopp's charges seemingly the most likely to deliver it.
After Milner drove a low shot straight at Gomes, the near telepathic link-up between Firmino and Coutinho saw the latter send his compatriot advancing through on goal.
From the kind of position that Firmino has so often scored from this calendar year, his decision to try a cute effort on the outside of his right foot saw him swipe harmlessly wide.
Yet Liverpool continued to probe for the opener and from more terrific combination play, Emre Can put Milner into space at the near post with a clever one-two.
Gomes was both quick off his line and brave to dive down at Milner's feet and deny him a shooting opportunity, though he could not prevent a corner.
And it was from the resulting set-piece that the breakthrough came. Jordan Henderson took the corner short to Milner, whose cross was ideally placed for Sadio Mane to glance a header over Gomes and into the far top corner.
As soon as it was 1-0, it was 2-0, as Liverpool doubled their advantage before the half-hour mark. The lively Coutinho, found by the forward ball of Firmino, made a yard of space and drove a powerful low shot which wrong-footed Gomes for his fifth league goal of the campaign.
That turned out to be Gomes' final contribution of the afternoon as an injury forced him off in serious discomfort, with Costel Pantilimon coming on for his league debut for the club.
Watford managed the game well for a period after that, but their spell of comfort only lasted for so long. Before half-time, Liverpool extended their lead to three goals from another flowing move.
Down the right Lallana was afforded too much time to watch the run of Can towards the far post, and his cross dropped for Can - who had escaped Amrabat - to nod in a header in the simplest fashion, his second goal in as many games.
Despite the scoreline, Watford weren't out of the game and they carried some danger going forward - but Karius had no trouble with Etienne Capoue's shot from the edge of the box.
And it was always Liverpool who looked the more likely of adding more goals. Can ghosted into space, perfectly found by Lallana's through ball, and looked for Firmino in space in the centre of the box but the forward couldn't get on the end of the low cross.
However, though they could arguably have enjoyed a greater winning margin at half-time, Liverpool still went into the break having scored three or more in the first-half of consecutive games since May 1998.
Liverpool add a couple more goals before the hour
The Reds' dominance continued in the second-half but it wasn't until just before the hour mark, that it paid off with more goals. While all of Liverpool's attacking class was once again on show, the suspect defending of Watford allowed Firmino a simple tap-in.
The Brazilian was in the right place at the right time for his fifth league goal this term, as Lallana earned himself a yard of space at the by-line and fired in a low cross which Firmino only had to get on the end of.
Once again, Liverpool struck in quick succession and added a fifth goal on the hour mark to match the size of Chelsea's win the previous day. Firmino ran on to Henderson's pass and briefly held it up before spotting Mane's run and putting in on a plate for the winger to score his second with an easy finish.
Watford fans cheered ironically when they won a corner with just over 20 minutes remaining, such was the manner in which their side were getting beaten, but they were nearly cheering for real a few moments later.
But Karius, who had only been asked to make a handful of testing saves in his previous appearance, was sharp to dive down to his left and parry Capoue's curling shot wide.
The German was called upon again soon after, forcing another corner as Miguel Britos' flick almost saw the ball sneak in at the near post.
Despite bringing on Sturridge and Wijnaldum, Watford were the more positive team late on and Karius had to make yet another good save, spreading himself well to thwart Troy Deeney with Milner on hand to clear the shot.
Wijnaldum wraps up proceedings after Janmaat consolation
And Watford's persistence was rewarded when Amrabat got the better of Milner in a duel, moved forward and fed Janmaat. The full-back's controlled left-footed finish nestled in to the bottom corner to deny Karius, and Liverpool, a clean sheet.
Klopp was visibly angry on the touchline and could have been left more seething, Karius preventing Odion Ighalo from making it 5-2.
Liverpool had opportunities to re-establish their five-goal lead but could not take them, substitute Sturridge twice hitting the woodwork and seeing a shot easily saved by Pantilimon and Coutinho firing over at the Kop end.
But the sixth goal finally came when Pantilimon could only beat Sturridge's shot in to the path of Wijnaldum, who was on hand to fire home a sixth.
With that, Liverpool moved to the top of the league table for the first time in well over two years and further confirmed that they are a team to keep an eye on this year.