The Costa Rica defender, in the side in place of the injured Leighton Baines, arrived unmarked at the far post to convert Romelu Lukaku's cross-shot in the 86th minute of an absorbing end-to-end contest.
It was a miserable first meeting with his former club for United manager David Moyes, who saw his side suffer a first defeat in 13 games in all competitions as they fell 12 points behind Premier League leaders Arsenal.
Wayne Rooney hit the base of the post with a deflected effort in the first half before Tim Howard made a stunning stop from Patrice Evra with Danny Welbeck heading the rebound against the bar in the second.
But Roberto Martinez's Everton defended resolutely throughout and snatched the points with time running out moments after Kevin Mirallas had rattled David De Gea's right-hand upright with a swerving free-kick from just outside the box.
If Moyes' transitional period has been difficult after leaving Everton for United, Martinez has slotted into the Goodison hot-seat.
Saturday's 4-0 win over Stoke took them above their hosts in the table and they started with the confidence of a team that has lost just once this season.
Lukaku was a menace, with even Nemanja Vidic finding his strength difficult to contain.
The Belgian threatened the home goal at regular intervals, bringing one excellent save out of De Gea after Vidic had been outmanoeuvred on the United left.
De Gea also denied Mirallas and, with Ross Barkley and James McCarthy also looking to profit from Lukaku's industry, at no stage were United comfortable.
It took them a good 15 minutes to exert any pressure, although after that they seemed to find space easy to come by without really doing much to exploit it.
Evra fizzed a cross into the six-yard area and Oviedo made a vital last-ditch tackle on Antonio Valencia.
Unsurprisingly though, United's most dangerous moments centred around Rooney.
The shot that cannoned back off a post had flicked off Phil Jagielka and Sylvain Distin before it reached its destination, and Howard did well to boot the rebound to safety before Shinji Kagawa could pounce.
His failure to turn home Kagawa's low cross shortly afterwards was a shock, even though it flew through a crowd of bodies, if only because of the form he has been in.
Given a second chance, his instinct was right as he hooked the ball goalwards but the power was missing and Everton managed to clear.
United began the second half with Evra launching a powerful drive goalwards.
However, with Everton again taking charge, the natives were getting restless and saw the introduction of Nani and Adnan Januzaj after less than an hour.
The move, which saw Valencia pushed into the right-back slot as Rafael and Kagawa made way, emphasised a lack of invention from United, although Nani blazing over not long afterwards was not exactly the intended response.
In contrast, Gerard Deulofeu almost made an immediate impact after replacing Barkley but again De Gea stood firm.
The flurry of activity that followed at the other end represented United's best opportunity of the contest.
But once Evra, Welbeck and Rooney had failed, the stage was set and Oviedo ensured Everton got the perfect finale.
To rub salt into the wounds, Rooney's booking after 61 minutes was his fifth of the season and he will now sit out the weekend clash with Newcastle.