Gylfi Sigurdsson's sublime late free-kick rescued a point for relegation threatened Swansea as United suffered two further injury blows.
Wayne Rooney had put the Reds ahead with a controversial penalty on the stroke of half time but the struggling Swans hit back to earn a full deserved point.
David de Gea was perhaps at fault for Sigurdsson's set piece strike and could have done better, but United were indebted to the keeper who pulled off a string of superb saves.
With a European semi final on Thursday, United's problems went from bad to worse when both Luke Shaw and an exhausted Eric Bailly were forced off through injury.
de Gea saved well from Fernando Llorente early on and Jesse Lingard then went close at the other end after Anthony Martial had picked him out.
Martial then fired straight at Lukasz Fabianski before Jordan Ayew was denied by de Gea having beaten the offside trap and burst clear.
Lingard fired wide from distance but United moved ahead in first half stoppage time, albeit in a controversial manner.
Marcus Rashford poked the ball beyond the advancing Fabianski and went down under the challenge from the keeper - even though replays suggested that contact was minimal to none.
Rooney scored from the spot for his second in as many matches to give the team a somewhat fortuitous half time lead.
United almost put the result beyond reach of the visitors on 52 minutes when Ander Herrera's cut back found Rooney ten yards out.
He looked certain to score but the ball inadvertently struck the back of Martial and rebounded away to safety.
Fabianski saved well from Martial before Bailly limped off clutching his knee to add to United's already-length injury list.
de Gea saved well from Sigurdsson before the Icelandic - looking to score at Old Trafford for the third season in a row - flashed a 20 yarder narrowly over the bar.
Swansea continued to press as de Gea was called into action again to keep out Llorente's close range effort with 20 minutes to play.
The Swans were level with 11 minutes to play, however, when Sigurdsson's wonderful 25 yard free-kick left de Gea rooted to the spot.
As good as the strike was, it was a good height for Dave and his failure to dive allowed Sigurdsson to place it perfectly into the corner.
Llorente went close with a header in the dying minutes, but once again this was a huge opportunity missed for the Reds to make headway on their top four rivals.
Overall team performance: 6/10
United Faithful Man of the Match: Anthony Martial. Troubled Swansea with his pace and energy.