Liverpool dominated in the early part and deservedly went 1-0 from the penalty spot through James Milner.
Iceland’s Gylfi Sigurdsson levelled up for Everton finishing off a swift country before £75m man Van Dijk decided the game with a header from a corner.
Liverpool had the game’s first attempt on goal through Sadio Mane who saw his long-range go wide after driving inside.
In the 25th minute Milner attempted a volley inside Everton box but his effort missed the far-post.
Liverpool eventually broke the deadlock on 35 minutes as Milner converted from the penalty spot after Mason Holgate fouled Adam Lallana.
Liverpool should have doubled their lead on 54 minutes only for an unmarked Joe Gomez failing to direct his header towards goal.
A minute later Lallana was played through on goal but hit his low strike was not enough to trouble Jordan Pickford.
Andrew Robertson went close for Liverpool in the 60th minute after beating his marker on the left only to see his left-foot shot superbly saved by Pickford.
Liverpool continued to create chances and should have added to their solitary strike but van Dijk’s header from a free-kick went straight to Pickford.
Everton who had struggled to create anything were back on level terms thanks to Sigurdsson who expertly slotted the ball beyond Loris Karius’ reach following Phil Jagielka’s cutback off a counter that also involved Ademola Lookman.
Roberto Firmino almost got on the score sheet after he was left unmarked inside the box from a corner but his volley was headed away by Jagielka.
Van Dijk put Liverpool 2-1 up in the 84th minute with a powerful header from Alex-Oxlade Chamberlain’s corner.
In the 90th minute Yannick Bolasie sent in a dangerous ball across Liverpool’s box off Lookman’s pass but Oumar Niasse could not connect.