Facing old club Everton for the first time, Moyes was handed plenty of stick by his former fans.
But what will cut Moyes even deeper is the fact that Roberto Martinez’s side now sit five points above his United team thanks to a player he bought as cover for Leighton Baines in 2012.
Costa Rican Oviedo was making just his third Premier League start for Everton but finished at the far post to secure Everton’s first win at Old Trafford since 1992.
Martinez had raised eyebrows when he suggested that he would end Everton’s inferiority complex at stadiums such as this and set out a 71-point target for Champions League football, but his team’s performance from the start gave that credence.
The Spaniard’s inference was clear enough and his Everton team, compact behind the ball and exploding forward in numbers, made United appear like the daunted rabbits many Blues sides looked like on occasions such as this.
James McCarthy and Gareth Barry were certainly stretched at times but, along with Ross Barkley, their high pressing game gave United a horrible start to the evening.
Romelu Lukaku has looked more polished than this but Nemanja Vidic never looked comfortable up against the powerful on-loan Chelsea man.
Lukaku missed with a powerful shot from just outside the area and had another deflected wide off Vidic when he was played through by McCarthy after Barkley had stolen the ball from Shinji Kagawa in midfield.
As United struggled to clear a corner from Steven Pienaar, David de Gea tipped over a thumping effort from Kevin Mirallas.
The hosts seemed shaken but they finally summoned some threat themselves when, with the visitors screaming for the ball to be kick out so Pienaar could receive treatment, Wayne Rooney had a powerful shot turned wide by Tim Howard.
Former United goalkeeper Howard showed his quality again when he pushed away a close-range stab from Kagawa and Moyes’ side went even closer when Ryan Giggs, playing for the first time since turning 40, glanced a header just wide from Antonio Valencia’s cross.
Giggs carved Everton’s defence open for the first time when he threaded a pass through for Rooney, whose shot glanced off Sylvain Distin and came back off the inside of Howard’s post.
De Gea was forced into more action when he pushed away from Lukaku before Barkley surged forward from midfield and just missed from more than 25 yards out.
After the break, United fed Patrice Evra into the left channel and the defender surged forward and hit a drive that was pushed away by Howard.
Marouane Fellaini’s struggle to justify his signing continued and an unconvincing header surrendered possession in a dangerous area before Barkley’s shot was deflected wide of goal by Chris Smalling.
As Fellaini attempted to atone for that, the ball looked like it would refuse to come down for him on the edge of the area before his half-volley was deflected wide of goal.
Moyes was moved to throw on Adnan Januzaj and Nani for Rafael and Kagawa as he looked to inspire his team but Nani lifted a shot well over the bar as the pattern of frustration continued. That was certainly clear when Rooney, who was fortunate not to be sent off in the 2-2 draw at Cardiff last month, attempted to kick out at Phil Jagielka.
He missed the Everton captain, who made out that he had been caught in the face, but was still given a yellow card.
Everton went close when Gerard Deulofeu was played in by Lukaku but could only come up with a disappointing shot straight at United keeper De Gea.
United broke down the other end straight away and Januzaj’s long-range effort was palmed over unconvincingly by Howard.
As the hosts sensed their chance, Rooney’s corner was met with a firm header from Evra, which was kept out by a fine save from Howard, before Danny Welbeck could only head on to the bar from a couple of yards.
Mirallas hit the outside of the post with a free-kick as Everton threatened but the Blues made the breakthrough with five minutes remaining.
Lukaku scuffed a shot but the ball reached Oviedo at the far post for the Costa Rican to stab in.
MANCHESTER UNITED (4-2-3-1): De Gea 7; Rafael 6 (Januzaj 58, 6), Vidic 6, Smalling 7, Evra 7; Fellaini 5, Giggs 7; Valencia 6, Kagawa 6 (Nani 58, 6), Welbeck 6 (Hernandez 81, 6); Rooney 7.
EVERTON (4-2-3-1): Howard 8; Coleman 6, Jagielka 7, Distin 6, Oviedo 6; McCarthy 7, Barry 7; Mirallas 6, Barkley 8 (Deulofeu 69, 7), Pienaar 6 (Osman 80, 6); Lukaku 7.