The Black Cats' unlikely victory means they are now out of reach of both Fulham, who lost 4-1 at Stoke, and Cardiff, who are bottom of the table after a 3-0 defeat at Newcastle.
The honeymoon period may already be over for United interim boss Ryan Giggs, who enjoyed a dream start to life in the dugout with a 4-0 win over Norwich last weekend.
Ten points from four games against Manchester City, Chelsea, Cardiff and now United mean Gus Poyet's men can only be caught by Norwich, who are three points back with Chelsea and Arsenal to play.
Giggs threw on Robin van Persie and Danny Welbeck in search of an equaliser in the second half but United suffered from the same lack of fluency that has cost them all season.
The defeat leaves United seventh in the table, 20 points behind league leaders Liverpool, while Sunderland know one more win will almost certainly be enough to survive.
Survival was always going to be an uphill task for Fulham who have been sitting in the league's bottom three since January 28.
Felix Magath's side knew only a victory would keep their hopes of safety alive but Peter Odemwingie, Marko Anautovic and Oussama Assaidi put Stoke three ahead in the second period.
Kieran Richardson pulled one back for the Cottagers with 10 minutes to play, but Jon Walters sealed the win and Fulham's relegation to the Championship.
Cardiff's stay in the top flight is also over as Newcastle recorded their first win in seven matches thanks to goals from Shola Ameobi, Loic Remy and Steven Taylor.
The Magpies had lost their previous six matches in a row and a number of the club's home fans walked out after 69 minutes to protest against the running of the club.
Aston Villa all but secured their Premier League status as they beat Hull 3-1 at Villa Park.
The win means Paul Lambert's side are now six points clear of the relegation zone with a significantly better goal difference than 18th-placed Norwich.
The hosts were ahead within a minute as Ashley Westwood converted Gabriel Agbonlahor's cross, but Hull levelled when Jordan Bowery scored an own-goal. An Andreas Weimann double, however, put Villa back in front and sealed a vital three points for the home side.
Rickie Lambert's 12th league goal of the season gave Southampton a 1-0 win at Swansea and means the Saints are now guaranteed to finish the campaign eighth in the table.
West Ham recorded a morale-boosting 2-0 victory over 10-man Tottenham to guarantee their Premier League safety for another season.
Spurs defender Younes Kaboul was sent off for bringing down Stewart Downing when the winger was clean through on goal and the Hammers took full advantage with goals from Andy Carroll and Stewart Downing.