Everton turn on former manager Moyes to beat United
Everton's Costa Rican midfielder Bryan Oviedo (L) scores past Manchester United's Spanish goalkeeper David de Gea (R) at Old Trafford in Manchester, northwest England on December 4, 2013
Wednesday's result left United 12 points adrift of leaders Arsenal and was the first time Everton, whom Moyes left at the end of last season to replace the retired Alex Ferguson, had won at Old Trafford since 1992.
Moyes endured a miserable evening, which began with him being jeered by his former supporters.
United went close to taking the lead when Wayne Rooney’s deflected shot came back off the post in the opening period before Danny Welbeck headed on to the bar from close range in the second half.
But Costa Rican Oviedo, signed by Moyes in 2012, turned in from close range to secure Everton's win with five minutes left.
Everton manager Roberto Martinez had promised that the Blues would show that they are shedding a supposed inferiority complex against the Premier League's top sides and from the start they were keen to prove his point.
Romelu Lukaku caused early panic in the United defence as the hosts scrambled to clear a cross from Kevin Mirallas under pressure from the striker.
Lukaku thumped wide from just outside the area and then had an effort deflected wide off Nemanja Vidic after Ross Barkley stole the ball from Shinji Kagawa in midfield.
United's defence, with Chris Smalling playing alongside Vidic at centre-back, struggled to get the ball away from the Steven Pienaar corner that followed before David de Gea was forced to tip over a powerful Mirallas shot.
The home side finally played their way into the match and a spell of pressure resulted in the ball falling for former Everton forward Rooney, whose drive was turned away by Tim Howard.
Former United goalkeeper Howard impressed again when he pushed away Kagawa's close-range effort.
And United went even closer to a goal when Antonio Valencia found space down the right and clipped over a cross that was glanced just wide by Ryan Giggs, playing for the first time since turning 40 last week.
Howard was beaten when Giggs picked out Rooney and the England striker's shot deflected off Sylvain Distin but the ball came back off the inside of the post.
Evra was in determined mood and broke beyond Mirallas to see a firm shot pushed away by Howard shortly after the interval.
A poor header from Marouane Fellaini against his old club gifted Everton possession in a dangerous area and United's Belgium midfielder was relieved to see Chris Smalling divert Barkley's shot wide of goal.
Moyes attempted to put some energy into his team when he threw on Nani and Adnan Januzaj for Kagawa and Rafael just before the hour.
But when the ball fell for Nani on the edge of the area the Portuguese lifted his shot well over the bar.
After being fortunate not to be sent off in the 2-2 draw at Cardiff last month, Rooney once again allowed his frustration to get the better of him, missing with an attempt to kick Phil Jagielka and earning a yellow card.
Everton substitute Gerard Deulofeu was played in by Lukaku but could only come up with a disappointing shot straight at de Gea.
And United broke down the other end of the field for Januzaj's long-range effort to be palmed over unconvincingly by Howard.
The home side were denied by the woodwork once again when Howard was at full stretch to keep out Evra's header from Rooney's corner before Welbeck headed on to the bar from close range.
Mirallas went close to putting Everton in front when his free-kick came back off the outside of the post.
But shortly afterwards, Everton got their breakthrough when Lukaku scuffed a shot to the far post and Oviedo turned in from six yards.