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Salah opens the scoring in first half
Mohamed Salah made the breakthrough in the 29th minute after Everton lost possession, with Cody Gakpo setting up the Egyptian forward, who calmly finished past Jordan Pickford to give Liverpool the lead.
Everton fight back after the break
Everton responded early in the second half when Beto bundled home from close range nine minutes after the restart, bringing the hosts level and energising the contest.
Van Dijk delivers stoppage-time decider
Just as the match appeared set to end in a draw, Van Dijk rose to meet a corner in stoppage time, heading home the decisive goal to snatch all three points for Liverpool.
League standings after derby win
The result leaves Liverpool fifth in the table on 55 points, three behind Aston Villa with five games remaining, while Everton sit 10th on 47 points.