Arsenal preserved their four-point lead by beating Hull City and United's cause was not helped by further victories for title rivals Chelsea, Manchester City, Tottenham Hotspur and Liverpool, for whom Luis Suarez scored four goals in a 5-1 thrashing of Norwich City.
Costa Rican left-back Bryan Oviedo scored the only goal of the game at a shellshocked Old Trafford in the 86th minute, to deal United manager David Moyes a bitter defeat in his first encounter with his former club.
Everton would not settle for a point, however, and after Kevin Mirallas had struck the post, Oviedo slammed home a low cross from Romelu Lukaku to condemn United to their fourth defeat of the campaign.
"I'm very disappointed," said Moyes. "We needed a bit of good fortune and we didn't get it. Everton kept at it and showed us what a good team they are. It's a big catch-up for us now."
Danish striker Nicklas Bendtner was handed his first Arsenal start since March 2011 and he made an immediate impact by opening the scoring in the second minute with a header from a Carl Jenkinson cross.
Mesut Ozil claimed the second goal early in the second half, ghosting onto Aaron Ramsey's neat pass and beating goalkeeper Allan McGregor at his near post.
Chelsea remain Arsenal's nearest challengers, but they had to come from behind to beat bottom club Sunderland 4-3 in a thrilling encounter at the Stadium of Light.
John O'Shea profited from some slack defending to draw Sunderland level early in the second period, but Hazard quickly restored the visitors' lead after a slick one-two with Lampard.
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