United went into the Old Trafford clash with Champions League qualification still in their hands, but an early injury to Luke Shaw added to Mourinho's defensive concerns with Marcos Rojo, Chris Smalling and Phil Jones already sidelined.
Wayne Rooney scored a controversial penalty at the end of the first half after Lucasz Fabianki was adjudged to have fouled Marcus Rashford, but United were dealt another setback as Eric Bailly - Mourinho's only fit centre-back - had to be replaced after the break with an ankle problem.
Swansea equalised through a fine Gylfi Sigurdsson free-kick to see United pick up their 10th draw at home this season, and hand Liverpool, Manchester City and Arsenal a boost in the race for the top four.
Mourinho told BBC Sport after the game: "We lost players and we lost points, so yes today was a bad day. We did not look tired and exhausted, we are tired and exhausted.
"You cannot isolate the performance out of the context. This is the ninth match of April, it is not human. We have a squad of 22 that is reduced to 13 or 14 players. The players are very tired.
"Did Jesse Lingard and Ashley Young look sharper than other people. Of course, they did not play the last match. I am more than happy with the players, they gave everything they could. I have zero criticism.
"I don't know about the injuries. 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.
"At the moment, we can walk from the bed to the toilet and break a leg."
Mourinho had called on the likes of defenders Jones and Smalling to show "crazy" commitment in order to get fit for the run-in, but chose to praise Juan Mata after the game after the Spanish playmaker's surprisingly quick return from groin surgery to make the bench.
"I prefer not to speak about Phil Jones and Chris Smalling, "he added. "I prefer to speak about Juan Mata giving everything to be available. I am grateful for that.
"We will give everything on Thursday. That is sure. Whether we go to the final or not doesn't matter, I am really happy with the guys."