Mane produced a pair of clinical finishes in the opening 20 minutes, the first a precise header into the top left corner and the second an outrageous flick with his back to goal.
Divock Origi added a third for the Reds midway through the second half with a firm strike inside the near post from the left to put the game beyond the visitors.
And Van Dijk doubled his personal Liverpool tally in the final 10 minutes by nodding in two headers at the Kop end from Trent Alexander-Arnold and Andy Robertson assists.
Jürgen Klopp’s side remained one point clear at the summit, with nearest rivals Manchester City 1-0 winners over West Ham United.
Team news
Roberto Firmino’s absence from the squad due to the ankle injury he suffered last weekend provided Origi with a chance in the XI, his third start of the season.
Alexander-Arnold came in at right-back, meanwhile, as James Milner moved into central midfield and Jordan Henderson dropped to the bench.
First half
With a vociferous Anfield crowd behind them, Liverpool started assertively and Milner’s high press almost caught out Ben Foster, the Watford goalkeeper’s clearance striking the Reds man but looping back into his hands.
Sheer quality, rather than luck, opened the scoring instead.
Alexander-Arnold’s swerving cross from the right corner of the area was a beauty, pitched perfectly over Craig Cathcart for Mane, whose flicked header showed the technique to match. The ball sailed in and the home supporters roared.
Klopp’s men were bursting with energy, forcing errors with their commitment to harrying the opposition every time they had the ball.
They came close to a second when Mohamed Salah picked out Alexander-Arnold’s run into the box on the overlap and the No.66 searched out Origi at the near post – but Watford cleared.
Liverpool – and Mane – had their second on 20 minutes, though.
Alexander-Arnold claimed another assist for the final cross, but Salah’s winning of a sprint with Adam Masina that covered nearly the entire Watford half took a large chunk of credit.
And yet the best was saved until last. Mane’s first touch took him away from goal, his second – a lofted backheel over Foster – doubled the advantage in impudent, exquisite fashion and set a personal best of 14 in a Premier League campaign for the forward.
This was the Reds at their relentless best and Salah, found by yet another pinpoint Alexander-Arnold pass, was denied by the woodwork 10 minutes before the break.
The Egyptian had rolled around his marker as he sped in from the right to collect the ball, but the eventual nudge at goal hit the near post and bounced away.
Liverpool did need a vital Joel Matip intervention to prevent Troy Deeney from tucking in Gerard Deulofeu’s cross, and the Hornets captain prodded wide from a carbon copy move moments later.
Second half
The pattern continued from the outset of the second half, though the visitors – winners of five away fixtures in their previous 11 games – retained a threat on the counter-attack.
Foster thwarted Salah after he had received Origi’s diagonal and scampered past Masina, then Matip was in the right place to block Cathcart’s punt having nodded out a Watford free-kick.
The Reds were eager for the third goal that would likely seal three more points and their play suggested it was going to come soon.
Mane threatened a hat-trick when he spread play out to the left and moved into the area to collect the return from Origi into the middle – but Cathcart deflected his strike into the Kop.
It did arrive on 66 minutes, however.
Origi supplied it, holding off the attentions of Daryl Janmaat as he cut in off the left and waiting for the right avenue to open before planting a finish between the near post and Foster’s reach.
Alisson Becker got down adeptly to push away substitute Andre Gray’s low drive from the angle and also touched a close-range effort from the same man over the bar within seconds, as Watford kept Liverpool honest.
But the final word of a fantastic display went to Van Dijk, the Dutchman a delighted scorer of two headers in front of the Kop, first courtesy of Alexander-Arnold’s free-kick from the right and the second via Robertson’s left-wing cross.