Sebastian Larsson's first-half goal was enough to give Sunderland their first win at Old Trafford since 1968 and take them to the brink of safety.
Barring an unlikely set of results, a single win from their remaining two games will be enough to preserve the Black Cats' top-flight status.
And it will be all but confirmed if Norwich lose on Sunday, after Sunderland condemned Manchester United to their seventh home league defeat of the season.
Norwich are three points adrift of Gus Poyet's men and must also overturn a 13-goal deficit in matches against Chelsea and Arsenal.
Sunderland, for whom this was a third successive victory, conclude their season with home games against West Bromwich Albion and Swansea.
But defending champions United have their own concerns.
Defeat brought interim United manager Ryan Giggs's winning start to an abrupt end, with Louis van
