The former Southampton star, who this month became the world’s most expensive defender, headed in the winner six minutes from time on a night not short on controversy at Anfield.
Having bemoaned the decision to award Everton a penalty in last month’s Premier League draw between the two, Jurgen Klopp saw his side get the benefit of some considerable doubt for another spot-kick call on Friday.
Mason Holgate was the man punished for a supposed foul on Adam Lallana just inside the area and James Milner stepped up to convert his first goal of the season.
Tempers flared before the interval and Holgate was again at the centre of it, shoving Roberto Firmino into the crowd to prompt an angry reaction from the Liverpool forward.
Sigurdsson then brought Everton level in the 67th minute, capping a fine breakaway to stun the home fans as the visitors had offered little in attack up to that point.
Sam Allardyce’s side, who confirmed the signing of striker Cenk Tosun at half-time, looked livelier after that but suffered the late blow of Van Dijk’s decider as Liverpool made it a club-record 16 games unbeaten against Everton.