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.