The 18-year-old - making his first appearance of the season - steered home Wayne Rooney's cross to spark wild celebrations on the away bench.
Moments earlier, David Meyler had blazed over in a rare Hull attack.
But it was a deserved win for United, who had gone close though Rooney, Zlatan Ibrahimovic and Paul Pogba.
United and Chelsea are now the only Premier League sides with a 100% record from three games, with Manchester City - who have two wins from two games - playing West Ham on Sunday.
Relive the late drama at the KCOM Stadium
Rashford's winner was the climax to a performance of persistence and intelligence from the visitors, who changed tack several times before eventually breaking down Hull.
Ibrahimovic arrived at Old Trafford with a reputation for audacity, but, in the first half, it was his physical size rather than mercurial skill that posed most threat.
Offering himself as a constant target in the box, the Swede headed just over the bar in the 10th minute before an extravagant backheel found the side-netting.
The excellent Curtis Davies blocked the more direct route to goal after the break, but the guile of Henrikh Mkhitaryan and the pace of fellow substitute Rashford stretched the Hull defence in different directions and, finally, beyond breaking point.
A back muscle injury which is likely to rule Marouane Fellaini out for several weeks means United will need to show similar adaptability when they meet Manchester City in their next match
With Harry Maguire returning from injury on the bench, former Manchester United assistant manager Mike Phelan had a season-high 14 fit senior professionals available for Hull.
But what his squad lacks in numbers, it makes up for in spirit.
Alongside Davies, Sam Clucas impressed in midfield, while Adama Diomande and Abel Hernandez chased hard to make something from the few scraps in attack.
Before Meyler's late wasted chance, Tom Huddlestone saw a shot deflect just wide with David De Gea hopelessly wrong-footed as they briefly threatened a smash-and-grab victory.
But, after wins over Leicester and Swansea in their opening two games, their 100% start to their top-flight campaign is over.
The uncertainty over the club's future might also be at an end, however, with vice-chairman Ehab Allam writing in the programme that the match could be his family's last as owners, with takeover talks with a Chinese consortium well advanced.
That deal may free up funds to supplement the Tigers' threadbare squad - Tottenham's Ryan Mason is a target - and help them continue to defy the Premier League's usual rules of gravity.
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Alan Shearer: "I don't think it will be long before he is a regular. It has to be absolutely brilliant for him at that football club, working, training and socialising with Zlatan Ibrahimovic and Wayne Rooney."
Ruud Gullit: "What I like about Rashford is that he goes straight past players. He gives you opportunities. I hope Manchester United manager Jose Mourinho considers him more because he's a great player."
Hull manager Mike Phelan: "You are playing quality sides who bring quality off the bench, but I am proud of the players. They have put hard work in today without getting anything out of it.
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"The Hull players are good players, and good people. They are determined and earned the right to be at this level. They worked hard for 12 months to be here.
"We have six points from nine and that is what dreams are made of with 13 players who are fit.
"We have been trying extremely hard to get players in and now have a small window. We have two or three in the pipeline but they are not done."
Manchester United manager Jose Mourinho: "We deserved it and in a more comfortable way. We had an amazing mentality. We tried and tried and got it.
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"I tell the boys every day we have to go into every match to win. We know we will draw or lose matches but the mentality has to be go to win. Today they showed that.
"Marcus Rashford gives new problems to the opposition so I am really happy with him and the team.
"We want to be champions, we don't want to start well, we want to finish well."
After the international break Hull travel to Burnley on 10 September with Manchester United at home to local rivals Manchester City on the same day.
Match ends, Hull City 0, Manchester United 1.
Second Half ends, Hull City 0, Manchester United 1.
Corner,  Manchester United. Conceded by Curtis Davies.
Substitution, Manchester United. Chris Smalling replaces Wayne Rooney.
Marcus Rashford (Manchester United) is shown the yellow card for excessive celebration.
Goal!  Hull City 0, Manchester United 1. Marcus Rashford (Manchester United) right footed shot from very close range to the centre of the goal. Assisted by Wayne Rooney.
Attempt missed. David Meyler (Hull City) left footed shot from outside the box is too high. Assisted by Adama Diomande.
Attempt missed. Paul Pogba (Manchester United) right footed shot from outside the box is too high. Assisted by Luke Shaw.
Attempt blocked. Luke Shaw (Manchester United) left footed shot from outside the box is blocked. Assisted by Henrikh Mkhitaryan.
Attempt saved. Henrikh Mkhitaryan (Manchester United) right footed shot from outside the box is saved in the centre of the goal. Assisted by Wayne Rooney.
Attempt missed. Paul Pogba (Manchester United) right footed shot from outside the box is close, but misses to the right. Assisted by Luke Shaw.
Attempt missed. Henrikh Mkhitaryan (Manchester United) left footed shot from outside the box misses to the left following a corner.
Corner,  Manchester United. Conceded by Jake Livermore.
Foul by Paul Pogba (Manchester United).
David Meyler (Hull City) wins a free kick in the defensive half.
Attempt missed. Marcus Rashford (Manchester United) header from the centre of the box is close, but misses to the right. Assisted by Wayne Rooney with a cross following a corner.
Corner,  Manchester United. Conceded by David Meyler.
Substitution, Hull City. Harry Maguire replaces Abel Hernández.
Corner,  Manchester United. Conceded by Eldin Jakupovic.
Attempt blocked. Henrikh Mkhitaryan (Manchester United) right footed shot from the centre of the box is blocked. Assisted by Marcus Rashford.
Attempt blocked. Zlatan Ibrahimovic (Manchester United) right footed shot from the left side of the box is blocked.
Corner,  Manchester United. Conceded by Eldin Jakupovic.
Attempt saved. Marcus Rashford (Manchester United) right footed shot from the left side of the box is saved in the bottom right corner. Assisted by Paul Pogba.
Attempt missed. Paul Pogba (Manchester United) right footed shot from outside the box misses to the right from a direct free kick.
David Meyler (Hull City) is shown the yellow card for a bad foul.
Henrikh Mkhitaryan (Manchester United) wins a free kick in the attacking half.
Foul by David Meyler (Hull City).
Attempt saved. Henrikh Mkhitaryan (Manchester United) left footed shot from outside the box is saved in the centre of the goal. Assisted by Antonio Valencia.
Wayne Rooney (Manchester United) is shown the yellow card.
Attempt missed. Tom Huddlestone (Hull City) right footed shot from outside the box is too high.
Corner,  Hull City. Conceded by Eric Bailly.
Attempt blocked. Tom Huddlestone (Hull City) left footed shot from outside the box is blocked. Assisted by David Meyler.
Attempt missed. Wayne Rooney (Manchester United) right footed shot from the left side of the box is close, but misses to the left.
Attempt blocked. Zlatan Ibrahimovic (Manchester United) right footed shot from outside the box is blocked.
Substitution, Manchester United. Marcus Rashford replaces Juan Mata.
Tom Huddlestone (Hull City) is shown the yellow card for a bad foul.
Henrikh Mkhitaryan (Manchester United) wins a free kick in the attacking half.
Foul by Tom Huddlestone (Hull City).
Attempt missed. Paul Pogba (Manchester United) right footed shot from outside the box is too high. Assisted by Luke Shaw.
Offside, Manchester United. Zlatan Ibrahimovic tries a through ball, but Juan Mata is caught offside.
Substitute Marcus Rashford scored an injury-time winner as Manchester United finally overhauled a stubborn Hull at a sodden KCOM Stadium.