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.
United's disappointment was compounded by the knowledge that they had spurned several chances to take the lead, with Wayne Rooney and Danny Welbeck hitting the woodwork and Ryan Giggs glancing a header narrowly wide.
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," Moyes said. "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."
Arsenal still have breathing space after a straightforward 2-0 win over Hull at the Emirates Stadium.
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.
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.
"Eden Hazard was excellent. He told me at dinner last night, 'Boss, for a long time I haven't scored.' He decided he would score tonight."
An own goal from City goalkeeper Costel Pantilimon and a late Victor Anichebe effort gave West Brom hope, but City held on.
The stand-out performance of the evening came at Anfield, where Suarez claimed his third hat-trick against Norwich in a magnificent individual performance.
He lobbed John Ruddy from 35 yards in the 15th minute, tapped home his second from a corner and then completed his hat-trick ten minutes before half-time with a crisp 18-yard half-volley.
His fourth goal, an inch-perfect 25-yard free-kick, took his tally of league goals to 13, and he even had time to set up a late goal for Raheem Sterling after Bradley Johnson had replied for Norwich.
Tottenham are three points back in sixth place, having spoilt Rene Meulensteen's first game as Fulham manager by coming from behind to win 2-1 at Craven Cottage.
Newcastle United's four-game winning run came to an abrupt halt in a 3-0 defeat at Swansea City, while Aston Villa overcame stubborn resistance to win 3-2 at Southampton. The game between Stoke City and Cardiff City finished goalless.