Gylfi Sigurdsson's brilliant free-kick put a dent in Manchester United's Champions League hopes as yet another Old Trafford draw gave relegation-threatened Swansea reason to believe in Premier League survival.
England striker Marcus Rashford's contentiously-won penalty just before half-time, converted by Wayne Rooney, had given United control of the match but Sigurdsson bent the ball home the 79th minute, seconds after Ander Herrera abandoned the post to join the wall.
It was a deflating way for United to set a new record unbeaten streak - 25 matches dating back to October 23 - and a more important statistic is the 14 league games they have drawn this season, 10 in front of their own fans.
And if that was not enough there was further bad news for United's overworked medical staff, with Luke Shaw and Eric Bailly picking up injuries to join a cluttered treatment room.
Mourinho, already missing a host of first-teamers, made four changes from Thursday's derby draw against Manchester City, with Rooney, Jesse Lingard, Ashley Young and Shaw recalled.
Shaw's third Premier League start of the year lasted just eight minutes, though, as he hobbled from the pitch in genuine discomfort.
For most of the half, United adopted the care-worn look of a squad stretched thin by congestion and circumstance.
There were uncharacteristic giveaways from Michael Carrick, miscued passes from Rooney and even a scream of frustration from the mild-mannered Rashford when an expected ball from Herrera was not forthcoming.
Anthony Martial was key to the two clear chances they did manage, floating the ball brilliantly over the top for Lingard, only for the England winger to fluff his connection after springing the offside trap.
The Frenchman went himself on the half-hour, cutting past two men from the left and unleashing a low drive which Lukasz Fabianski almost spilled at Lingard's feet.
For their part, Swansea were bright and up for the contest.
Fernando Llorente, collecting a smart pass from Tom Carroll, had David De Gea flapping when he turned Bailly at the back post and let fly. Andre Ayew and Sigurdsson both drew further saves from the Spaniard but United took the lead when Rashford made his controversial intervention.
Having sprinted clear he converged with Fabianski, though the teenager was well on his way to the ground when Fabianski touched him. Rooney, for 252nd time, did the rest for United.
He should have had a second early in the second half, but hit Martial's back rather than the target from 10 yards.
Bailly pulled up around the hour mark, leading to a second reshuffle of the red defence and giving Swansea renewed hope.
Rooney's foul on Martin Olsson gave them the opening they needed and Sigurdsson lined up from 20 yards.
His strike was clean and curled perfectly inside the post but would have hit Herrera square in the chest had he not sprinted away from his station moments earlier.
Either side might have won it in an ill-tempered finale, Lingard heading straight at Fabianski at the far post and Llorente failing to meet a glorious cross in added time.