The Reds’ £34 million summer signing from Southampton reacted first to a loose ball for the 1-0 Premier League win.
This was after substitute Daniel Sturridge’s shot had hit the post in the fourth of eight minutes of stoppage time.
Everton, who had lost goalkeeper Maarten Stekelenburg to injury in the second half, have now won just one of their past 20 meetings with their local rivals.
Liverpool — who moved up to second on the league table — created the better chances, with Stekelenburg and his replacement Joel Robles saving well from Roberto Firmino.
There was a flashpoint in the second half when a dreadful tackle by Ross Barkley on Jordan Henderson sparked angry scenes.
The Everton midfielder was fortunate to escape with only a yellow card from referee Mike Dean.