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Arsenal's defeat, their first in 12 league games, sees them remain second, two points behind City and level on points with third-placed Liverpool with six games left to play.
Bundesliga leaders Bayer Leverkusen secured a last-gasp 2-0 home win over West Ham United.
Juergen Klopp's side have 70 points from 30 games, two ahead of Arsenal and three more than champions Manchester City.
Liverpool are second with 64 points, behind the Gunners on goal difference, with 10 games remaining of a thrilling three-way title race..
Van Dijk rose high above his marker to head a corner back across goal and into the far corner in the 118th minute.
Liverpool's patched-up side came out after the break in top gear.
South Korea's exit from the Asian Cup following a 2-0 loss to Jordan in the semifinals this month, which extended their 64-year wait for a third title, led to calls from fans and some politicians for the 59-year-old's dismissal.
Liverpool won 3-1 while Manchester City beat Everton 2-0 in an earlier match.
"Congratulations to Arsenal. We were not good enough today," Klopp said. "It just shows the boys are human beings."
Liverpool moved five points clear atop the Premier League to set up a potential thriller on Sunday against third-placed Arsenal.
