Marouane Fellaini scored a 78th-minute winner as United beat Crystal Palace 2-1 to move provisionally seven points clear of fifth-place Liverpool. Anything other than a win for Liverpool at champion Chelsea on Sunday and United is back in the Champions League after a one-year absence.
"We have made a very important step," United manager Louis van Gaal said.
The rest of the focus Saturday was on the fight to avoid relegation, which proved beyond Burnley despite a 1-0 win at Hull.
Wins for Sunderland, Leicester, and Aston Villa pushed them clear of the bottom three, and left Hull and Queens Park Rangers as the teams most likely to go down with Burnley.
"We've had a go, but in the big moments in the important games it hasn't gone for us," said Burnley manager Sean Dyche, who largely kept faith with the players who earned promotion last season to protect the financial security of the club.
Last-place QPR is nine points from safety with three games left, the first of which is away at Manchester City on Sunday when relegation could be sealed. Hull is third from bottom and two points from its nearest rival, Newcastle.
"We have got a mountain to climb," said Hull manager Steve Bruce, whose team finishes the season with games against Tottenham away and Man United at home.
If Liverpool comes away from Stamford Bridge with a win or a draw, United has two chances to clinch a return to the Champions League — first against Arsenal and then at Hull.
Consecutive losses to Chelsea, Everton, and West Bromwich Albion jeopardized United's top-four hopes and the win at Palace was hard-fought, with goalkeeper David de Gea producing crucial saves either side of Fellaini's winning goal.
Juan Mata's 19th-minute penalty for United was cancelled out by a curling free kick from Jason Puncheon that flicked off the head of United's Daley Blind to find the net in the 57th.
Against the run of play, Ashley Young cut inside and sent over a deep cross that Fellaini met with a downward header as Palace's goalkeeper and defenders got in each other's way.
Burnley picked up many admirers in its one season back in the Premier League for its work rate and devotion to positive football — there was even a famous win over Manchester City and a draw at champion Chelsea — but a scoring record of 27 goals in 36 games proved its undoing.
Danny Ings bagged the team's first goal in seven games for the winner against Hull, but it was in vain.
"To say they have been written off from the start by so many . their resilience has been fantastic," Burnley manager Sean Dyche said of his players. "We did what we had to do today but we have not done it enough over the course of the season. Scoring goals has been our problem."
Saturday's results left five teams in the scrap to avoid the third relegation spot — most pundits are giving QPR no chance of a miraculous escape — but Hull is now most in danger.
Sunderland won 2-0 at Everton thanks to second-half goals by Danny Graham and Jermain Defoe to climb out of the bottom three. Aston Villa beat West Ham 1-0 at home thanks to Tom Cleverley's third goal in as many games and is four points clear of safety. Leicester beat Southampton 2-0 courtesy of a double by Riyad Mahrez, with a sixth win in seven games pushing the team three points clear of trouble.
Newcastle drew 1-1 at home to West Brom — ending an eight-match losing run — and that point could yet be enough. West Brom and Palace are assured of another season in England's lucrative top league.
Tottenham's slim chances of securing a Champions League spot ended with a 3-0 loss at Stoke, continuing a late-season slump.
Charlie Adam and Steven Nzonzi gave Stoke a 2-0 lead by halftime and Tottenham's day got worse when Vlad Chiriches was sent off for a second booking in the 51st. Mame Biram Diouf scored the third.