The £75 million signing had an impressive debut which was crowned with a fine headed goal from an Oxlade-Chamberlain corner late in the game where James Milner had opened Liverpool's account with a well taken penalty, in the first half, in which Liverpool looked comfortable.
Everton's doggedness and at times enterprising football was finally rewarded on a rare breakaway when Glyffi Sigurdsson scored in the 67th minute with a shot from the edge of the box to beat Karius.
Although Liverpool were the better team all evening, Everton were right back in the game.
A succession of Liverpool corners, all well-delivered by Oxlade-Chamberlain came close until the 86th minute when the £75 million man Van Dijk beat Jordan Pickford to the ball and directed his header into the Everton net for a worthy lead for the Reds.
Liverpool held on from there and it ended in a nervy but satisfactory FA Cup victory against their great rivals.