A run of three draws in four league matches had seen Liverpool's advantage over Manchester City trimmed to a point heading into Wednesday's fixtures.
However, Liverpool never looked like a team in danger of cracking under the pressure as Mane's brilliant brace - highlighted by a stunning backheeled finish for his second - put them in command.
Divock Origi put the result beyond doubt in the second half and a pair of late headers from Van Dijk completed the rout.
City remain firmly on their heels, however, the defending champions beating West Ham 1-0 to maintain an extremely slender gap at the league's summit.
Watford came to Anfield on the back of a 5-1 win at Cardiff City but any fears of them denting Liverpool's challenge were calmed in the ninth minute, the Hornets punished for poor defending out wide and in the box as Trent Alexander-Arnold whipped in a superb cross for the unmarked Mane to rise and head into the top-left corner.
That same pair combined again on 20 minutes as Liverpool threatened to run riot. The ball was eventually worked to Alexander-Arnold after a surging run from Mohamed Salah and the England full-back again found Mane, who made up for a poor first touch in spectacular style by beating a helpless Ben Foster with a gloriously audacious backheel.
Only the woodwork prevented Salah from making it three from a tight angle, before Troy Deeney wasted a gilt-edged chance to halve the deficit as he failed to connect cleanly with a Gerard Deulofeu cross.
Liverpool took their foot of the gas somewhat after the restart, but Origi extinguished any lingering hope of a Watford comeback by cutting in from the right and powering in his first league goal since a Merseyside derby winner in December.
Alisson made a pair of fine saves from Andre Gray before Watford's misery was compounded, Van Dijk heading home twice in the space of three minutes to round things off.
Reds return to ruthless ways
Liverpool had been kept at bay by Bayern Munich and Manchester United with relatively little difficulty in successive dour 0-0 draws, but this was a return to the kind of form that has defined their title challenge. Their display will have been a welcome sight to Jurgen Klopp, who cut a frustrated figure after the Old Trafford stalemate.
