The 1-0 victory for Louis van Gaal's team over neighbor Manchester City at the Etihad Stadium boosted its hopes of winning qualification for the Champions League next season.
City still occupies the fourth and final spot in the English Premier League, but West Ham and United in sixth have closed to within a single point with eight games to play.
For Manuel Pellegrini's side a damaging fifth home defeat has ended any hopes of mounting a title challenge and injuries in each half to England internationals Raheem Sterling and goalkeeper Joe Hart added to the headaches.
When Sterling limped off with a groin injury his side was already trailing to a piece of impudent skill from 18-year-old Rashford.
He ghosted past an ineffectual challenge from Martin Demichelis and beat Hart with a low shot.
Visitor United edged the first half and had strong penalty claims turned down near the interval when Rashford was clumsily bundled over by Demichelis.
The defender's uncomfortable afternoon continued after the break and his under hit back pass interested Anthony Martial, with Hart hacking the ball away first.
But in doing so, he injured his calf and was stretchered off with League Cup final penalty hero Willy Caballero taking over in goal.
He had to be smart to push away a Martial effort with Rashford nearly scoring from the rebound.
City worked up a head of steam and Sergio Aguero hit the outside of the post with a headed effort before being involved as substitute Wilfried Bony pulled his shot wide.
Aguero might have leveled in injury time, but United held on for a deserved victory, just three days after going out to arch rival Liverpool in the last 16 of the Europa League.
Tottenham keeps pace at top
The other late kickoff Sunday saw Tottenham Hotspur maintain its title challenge with a comfortable 3-0 home win over Bournemouth.
EPL top scorer England Harry Kane put Spurs ahead with two first half goals, the first after just 44 seconds, the second set up by young midfield star Dele Alli, to take his league tally to 21.
Kane's second half shot was only parried by Artur Boruc and Christian Eriksen scored on the rebound.
The win leaves Tottenham five points adrift of leader Leicester, with both having seven games to play.
Liverpool, so impressive in knocking out Manchester United, slumped to a 3-2 defeat at Southampton after building a two-goal first half lead through goals from Philippe Coutinho and Daniel Sturridge.
But a Sadio Mane double, with a goal from Graziano Pelle sandwiched between, gave Ronald Koeman's team an unlikely win.
Mane could even afford to have a penalty saved at the start of the half by Simon Mignolet before the fightback began in earnest.
The early kick off saw Tyne-Wear derby rivals Newcastle and Sunderland battle to a 1-1 draw which left both teams in the relegation zone.
Sunderland looked headed for a seventh straight victory over Newcastle after Jermain Defoe's clever first half volley.
But Newcastle gave its new manager Rafa Benitez an early boost when Aleksandar Mitrovic leveled with seven minutes remaining at St James' Park.
Barca held in El Madrigal
In La Liga Sunday, Barcelona conceded a two-goal lead at Villarreal, but still extended its unbeaten run in all competitions to 39 games in the 2-2 draw.
First-half goals from Ivan Rakitic and Neymar at El Madrigal looked set to send Barca 11 points clear, but Villarreal, hoping for a top four slot, hit back after the break.
Cedric Bakambu cut the deficit in the 57th minute, before an own-goal from Jeremy Mathieu made it 2-2 six minutes later.
Both sides might have edged it in a pulsating affair, but honors were even at the end, Barca still stretching its lead to nine points over Atletico Madrid, which lost 2-1 at Sporting Gijon Saturday.
Real Madrid is a further point back after a later impressive 4-0 win over Sevilla with Karim Benzema, Cristiano Ronaldo, Gareth Bale and Jese Rodriguez on target. Real and Barcelona meet in El Clasico on Saturday April 2.
Buffon Serie A record
In Italy, there was a new Serie A record for stalwart Juventus goalkeeper Gianluigi Buffon in a 4-1 derby win for the reigning champion over Torino.
Juve was leading 2-0 when Buffon was beaten from the penalty spot by Andrea Belotti, the first goal he had conceded in 973 minutes, beating Sebastiano Rossi's record of 929 minutes for AC Milan in 1994.
"I love playing at this club and so this is a record for all of Juventus, not just me," the 38-year-old Buffon said.
His side's 22nd win of the season kept it three points clear of Napoli, which beat Genoa 3-1 later Sunday with Gonzalo Higuain scoring twice to take his Serie A goals tally to 29 in 30 games.
It was also some consolation for the heartbreaking Champions League defeat to Bayern Munich in midweek after building a two-goal aggregate advantage.