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," 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."
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.
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.
Jozy Altidore gave the home side a 14th-minute lead, only for goals from Frank Lampard and Eden Hazard to put Chelsea in front at half-time.
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.
The game's final stages belonged to Sunderland full-back Phil Bardsley, who scored an own goal in the 84th minute before quickly atoning by scoring at the other end two minutes later.
"I know I will be close to a heart attack sometimes (with Chelsea)," said Chelsea manager Jose Mourinho.
"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."
City are two points behind Chelsea after winning 3-2 at West Bromwich Albion in a game delayed by 15 minutes due to traffic congestion.
Manuel Pellegrini's side -- for whom captain Vincent Kompany made his first start since October 5 -- went 3-0 up through Sergio Aguero and a Yaya Toure brace.
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 10 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.
"Luis has got to be up there as one of the best strikers in the world after that," said Liverpool manager Brendan Rodgers, who had seen his side beaten 3-1 by Hull on Sunday.
"The repertoire he showed here tonight was incredible."
The result left Liverpool in fourth place, a point shy of City and above Everton on goal difference.
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.
Ashkan Dejagah put the hosts head in the 57th minute, but Vlad Chiriches and Lewis Holtby scored in the last 17 minutes to ease the pressure on Spurs manager Andre Villas-Boas.
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.