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Kyle Walker marked his 100th Premier League appearance for Manchester City with a goal against his former club.
The right-back, who came through the ranks at the Blades, struck midway through the first half with a superb long-range strike.
It was a deserved goal for a Manchester City side who dominated the first half, and they could have added another but for an excellent save by Aaron Ramsdale from Ferran Torres' header.
Ramsdale did well again in the second half when he dived at full stretch to keep out a Riyad Mahrez free-kick.
But from then on the visitors struggled to find a second goal that would kill the game off and United should have capitalised when substitute John Lundstram fired just over from Sander Berge's cross.
The victory moves Man City into seventh ahead of the remaining weekend's games and they are now unbeaten in seven matches across all competitions.