Questions were starting to be asked after four draws in their last five matches in all competitions but this performance was a return to the verve and inventiveness which they desperately needed to rediscover.
It was typified by Mane's – and Liverpool's – second of the night: a brilliant backheel from 12 yards out which caught goalkeeper Ben Foster unawares.
The Senegal international had already, somewhat surprisingly, scored with a header having been given the job of central striker in the absence of the injured Roberto Firmino.
Firmino's replacement Divock Origi, who was posted on the left wing to accommodate Mane in his new role, added the third in the second half with Virgil Van Dijk scoring two headers in the final 11 minutes.
Mane's double meant this is his best Premier League goalscoring return of 14 goals, with 10 matches still to go, and was the fifth home league match in succession in which he had found the net.
The result was exactly the boost Liverpool, still a point ahead of Manchester City, needed to inject life into what some argued was a flagging title challenge and the last thing Everton, hosts of Sunday's Merseyside derby, wanted to see.
And it was hardly as if Watford were there for the taking, arriving at Anfield having won six and drawn four of their last 11 matches.
But they ran into a re-energised and clinical Liverpool side who took out their frustrations from Sunday's goalless draw at Old Trafford on opponents who were just not allowed into the game.
The move of playing Mane through the middle may have been more down to necessity than design but it paid off with the Senegal international providing the cutting edge which had been missing.
Trent Alexander-Arnold's return at right-back was also a key factor as he and Mohamed Salah ripped Watford's left flank to shreds with Adam Masina enduring a torturous evening.
The 25-year-old received little in the way of support from midfielder Roberto Pereya in front of him and it was no surprise three of the goals came from that side.
Liverpool's opener, after nine minutes, was simple in its execution as Alexander-Arnold whipped over a cross so perfect Mane, the smallest man in the penalty area, scored a classic centre-forward goal as he rose to head home.
Mane's second came from the same area with Salah cutting Watford to shreds down the right allowing Alexander-Arnold to swing over another inch-perfect ball.
The striker's first touch took him away from goal but the 26-year-old showed an awareness not always associated with him in the final third, surprising Ben Foster with a backheel from 12 yards out.
Salah, who was denied by a combination of Foster and a post, was the architect of the move of the match when he collected his raking crossfield ball back from Origi and dinked a chip through for the on-rushing Andy Robertson to volley wide.
After the break Mane was denied a hat-trick by Craig Cathcart's close-range block but Origi got the goal his considerable leg-work down the left deserved when he cut in to beat Foster at his near post in the 66th minute.
Alisson Becker twice denied substitute Andre Gray before Van Dijk struck with two headers from crosses by the superb Alexander-Arnold and fellow full-back Robertson.