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The heavy defeat leaves Everton 16th on 27 points, two places and two points above the relegation zone.
Juergen Klopp's side have 70 points from 30 games, two ahead of Arsenal and three more than champions Manchester City.
Van Dijk rose high above his marker to head a corner back across goal and into the far corner in the 118th minute.
City tested Chelsea's back line with 31 shots to the visitors' nine, but only five were on target.
Liverpool moved five points clear atop the Premier League to set up a potential thriller on Sunday against third-placed Arsenal.
Leon Bailey's deflected strike after a surging run in the 74th minute sealed Villa's club record-equalling 14th successive home league win as they moved above City into third place.
Cole Palmer scored a late penalty to make it 4-4 at Stamford Bridge in London.
It was a triumphant return to Spurs for Mauricio Pochettino who got the biggest win of his short Chelsea tenure.
Victory pushed Arsenal up to second place with 24 points from 10 games, two behind Tottenham Hotspur.
Chelsea squandered a two-goal advantage to end up with a point.