The hosts dominated the match but could not find a way past the Tigers goalkeeper, who brilliantly kept out Zlatan Ibrahimovic's long-range strike and Paul Pogba's driving effort in the first half.
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In between, Harry Maguire should have done better with a header which he put wide of goal.
Ibrahimovic hooked an effort wide in the second half and Jakupovic made his best save to prevent Juan Mata from scoring at the back post, as well as keeping out Paul Pogba's curler.
The visitors could have won it with five minutes to go, but on-loan Lazar Markovic's clipped shot came back off the post and Abel Hernandez struck tamely at David de Gea.
The point keeps United in sixth place, but allowed Hull to move off the bottom of the table.
The rules are different for me - Mourinho
Relive the entertaining draw from Old Trafford
Jakupovic made a total of six saves, punching the air in delight with each effort he kept out and taking the acclaim of the jubilant away supporters at full-time.
Hull have shipped 47 goals this season - only Swansea (52) have conceded more in the division - and this was just their second clean sheet in 23 league games.
Asked by BBC Sport if it was his best game in a Hull shirt, Jakupovic replied: "I try to be my best for the team all the time but today I caught a good day.
"The striker celebrates when he scored, and I celebrated to myself with some saves."
United striker Ibrahimovic was not impressed by the Hull player's performance. The Swede said: "I did not see any chances where it was difficult for the goalkeeper. It was not a good save from Mata, it was a bad finish. Some saves he made for the cameras."
United had seen all the top four sides drop points in this round of fixtures as they chase a Champions League spot, but failed to capitalise even though they had 66% possession in the match.
Despite extending their run to 14 games unbeaten in the top-flight, they have drawn their last three games and are four points adrift of Liverpool in fourth place.
United only had themselves to blame in a wasteful performance. Marcus Rashford, who completed a full 90 minutes for the first time since November, highlighted his team's sloppiness by losing possession 21 times - more than any other player on the pitch.
Wayne Rooney was brought off the bench at half time, but failed to change the game, having become the club's leading all-time goal scorer in the previous league match at Stoke.
Manchester United manager Jose Mourinho: "We didn't score. You don't score, it is not possible to win.
"Their goalkeeper was good.
"We needed to score, we needed more time to play. If you played 35-40 minutes in both halves, it is a lot. I think Hull City tried to see where they could go, the way they could behave and tried to see what the referee would allow them to do.
"They had the feedback and were comfortable to do what they did. I am not critical of that. They are fighting against relegation and every point is gold.
Asked by BBC commentator Martin Fisher what upset him about referee Mike Jones' performance: "If you do not know football, you should not have a microphone in your hand."
Before this game, Hull had lost nine straight away games, with their last point on their travels coming at Burnley in early September.
But under new boss Marco Silva they have shown enough improvement to suggest they can preserve their top-flight status.
The Portuguese has led Hull to a win and a draw in his first three games - with a defeat coming against leaders Chelsea - and lie four points away from safety.
Having beaten United in the second leg of their EFL Cup semi-final last week, Hull may even feel disappointed by not taking all three points with Markovic coming agonisingly close to clinching the winner late on.
However, striker Oumar Niasse was lucky not to be given a red card after making late challenges on Michael Carrick and Daley Blind, having earlier received a yellow card.
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Hull boss Marco Silva: "It is a very good result for us against a very good team. We played like a team with great attitude, spirit and character. What we showed tonight again, I am happy.
"Sometimes you have to suffer in moments but we have to play as a team.
United travel to champions Leicester City on Sunday (kick-off 16:00 GMT), while Hull host title challengers Liverpool on Saturday (15:00 GMT).
Match ends, Manchester United 0, Hull City 0.
Second Half ends, Manchester United 0, Hull City 0.
Attempt blocked. Paul Pogba (Manchester United) right footed shot from outside the box is blocked. Assisted by Daley Blind with a headed pass.
Attempt saved. Paul Pogba (Manchester United) right footed shot from the left side of the box is saved in the top right corner. Assisted by Daley Blind with a headed pass.
Attempt saved. Marcos Rojo (Manchester United) header from the centre of the box is saved in the top left corner. Assisted by Paul Pogba with a headed pass.
Juan Mata (Manchester United) wins a free kick in the attacking half.
Foul by Sam Clucas (Hull City).
Foul by Zlatan Ibrahimovic (Manchester United).
Eldin Jakupovic (Hull City) wins a free kick in the defensive half.
Delay over. They are ready to continue.
Delay in match Andrea Ranocchia (Hull City) because of an injury.
Attempt saved. Abel Hernández (Hull City) left footed shot from outside the box is saved in the bottom left corner. Assisted by Tom Huddlestone.
Marcus Rashford (Manchester United) wins a free kick on the left wing.
Foul by Omar Elabdellaoui (Hull City).
David Meyler (Hull City) is shown the yellow card.
Paul Pogba (Manchester United) wins a free kick in the defensive half.
Foul by David Meyler (Hull City).
Foul by Zlatan Ibrahimovic (Manchester United).
Harry Maguire (Hull City) wins a free kick in the defensive half.
Foul by Antonio Valencia (Manchester United).
Sam Clucas (Hull City) wins a free kick in the defensive half.
Lazar Markovic (Hull City) hits the left post with a right footed shot from the left side of the box. Assisted by Tom Huddlestone.
Delay over. They are ready to continue.
Delay in match Sam Clucas (Hull City) because of an injury.
Wayne Rooney (Manchester United) is shown the yellow card.
Foul by Ander Herrera (Manchester United).
Sam Clucas (Hull City) wins a free kick in the defensive half.
Attempt blocked. Ander Herrera (Manchester United) right footed shot from outside the box is blocked. Assisted by Antonio Valencia.
Offside, Manchester United. Daley Blind tries a through ball, but Zlatan Ibrahimovic is caught offside.
Lazar Markovic (Hull City) is shown the yellow card for a bad foul.
Hand ball by Zlatan Ibrahimovic (Manchester United).
Sam Clucas (Hull City) is shown the yellow card.
Wayne Rooney (Manchester United) wins a free kick in the attacking half.
Foul by Tom Huddlestone (Hull City).
Antonio Valencia (Manchester United) wins a free kick on the right wing.
Foul by Lazar Markovic (Hull City).
Corner,  Manchester United. Conceded by Tom Huddlestone.
Attempt blocked. Paul Pogba (Manchester United) right footed shot from outside the box is blocked. Assisted by Juan Mata.
Wayne Rooney (Manchester United) wins a free kick in the defensive half.
Foul by Tom Huddlestone (Hull City).
Eldin Jakupovic made a string of fine saves as Hull frustrated Manchester United by claiming a goalless draw in the Premier League at Old Trafford.