Liverpool scored a goal in each half as City's winless run was extended at Anfield.
The Blues fell behind on 12 minutes to a Cody Gapko goal and withstood further pressure from the home side, but dug in and went into break trailing by just the one goal.
A much more cohesive second-display saw City grow into the game, but the failure to create any clearcut opportunities was eventually punished by a Mo Salah penalty on 78 minutes.
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