Jose Mourinho's men also have the chance to qualify for the Champions League by winning the Europa League, with a trip to Celta Vigo up next for the first leg of their semi-final on Thursday.
Eric Bailly was also forced off against Swansea and Mourinho's Reds are still reeling from Marcos Rojo's season-ending knee injury.
"We are exhausted", United manager Jose Mourinho said.
"I think we really turned a corner last week and showed that again today". I prefer to speak about (unused substitute) Juan Mata giving everything to be available.
It was another frustrating day for Mourinho, even though his United side extended their unbeaten Premier League run to a club record 25 games.More news: Alonso's Indy 500 car will go to his museum
The Portuguese boss has allegedly told the first-team he doesn't want them to publish any training pictures within 48 hours leading up to a game, especially from the team bus whilst on the way to games.
But Mourinho has grown increasingly frustrated with the tendency of his players to turn to social media, particularly when recovering from high-profile injuries.
"We did not look tired and exhausted, we are tired and exhausted", he insisted.
Manchester United manager Jose Mourinho said: "It's very hard, I can not isolate the performance to the context".
After nine games in April there is no let up between now and the end of the campaign, with United facing games every three or four days.More news: Rohit Sharma, Shami Return To Indian Squad For Champions Trophy
'I think Luke Shaw's must be a big injury, because to leave the pitch after 10 minutes I am expecting a very big injury.
Mourinho feels his squad is simply paying the price for a hectic fixture schedule but would not blame those who started the match as they slipped to a 10th home league draw of the season. His only recognisable defenders are Antonio Valencia, Daley Blind and Matteo Darmian.
"It doesn't matter if I go with (Matteo) Darmian central defender, if I go with (Michael) Carrick or Axel (Tuanzebe), I trust the boys, the spirit is fantastic, the group is phenomenal".
United were 1-0 up against relegation strugglers Swansea on Sunday but had to settle for a point when Gylfi Sigurdsson equalised with a free-kick 11 minutes from time.
Against Swansea, Lingard did little to impress with most of the creative spark coming from his fellow attackers Anthony Martial and Marcus Rashford. "I'm using the word 'deceived.'".More news: Pentagon chief: USA studying Moscow's Syria safe zone plan
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