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In a high tempo, if often scrappy, game Ole Gunnar Solskjaer's United side made the most of their chances while Chelsea, whose form at home has been suspect all season, squandered a string of opportunities.
A fine glancing header from Anthony Martial just before halftime from an Aaron Wan-Bissaka cross put the visitors ahead. It was United's first Premier League goal in four games.
Defender Harry Maguire made it two in the 66th minute, losing marker Antonio Rudiger to head home from a corner by United's new Portuguese signing Bruno Fernandes.
Chelsea had two goals ruled out by VAR. Kurt Zouma scored from a corner in the 54th but there was a push in the area. Substitute striker Olivier Giroud's headed effort in the 77th was cancelled when he was adjudged marginally offside.
The loss left Chelsea fourth, a point clear of resurgent rivals Tottenham Hotspur led by their ex-coach Jose Mourinho. Chelsea host Spurs on Saturday. United, another of Mourinho's old clubs rose to seventh, three points behind Chelsea.