At the end of a disappointing first period, Marcus Rashford won a controversial penalty for the hosts after going down under a challenge from Lukasz Fabianski. Wayne Rooney converted the opportunity in first-half stoppage time.
Swansea came on strong in the second period, though, and Sigurdsson curled home a sublime free-kick in the 79th minute to put the visitors level.
United have suffered with a succession of injuries recently, so it was a huge blow for the Red Devils to see Luke Shaw hobble off early on. He was replaced by Antonio Valencia.
As has often been the case at Old Trafford this season, the home side found it difficult to get out of first gear to begin with.
Fernando Llorente stung the palms of David De Gea with a sharp shot, and Sigurdsson was inches away from turning home a rebound after the United goalkeeper parried Jordan Ayew's low strike. Per football journalist Andy Mitten, the hosts were struggling:
Swansea's defensive frailties meant there were still some opportunities for United, with Jesse Lingard latching on to Anthony Martial's dinked pass. The forward saw his effort stopped by Lukasz Fabianski.
There were small signs towards the end of the half that the hosts were growing into the game. After their best spell of passing in the match, Fabianski felled Rashford in the area. Three minutes into first-half stoppage time, Rooney stepped up and calmly finished.
Per Nooruddean Choudry of Joe.co.uk, the decision was harsh on Swansea, with the young forward appearing to take a dive:
After half-time, Rooney had an excellent chance to double his side's advantage when Ander Herrera cut the ball back into the area. But the Red Devils captain could only fire at the back of his own team-mate.
In another blow to the Red Devils, another defender went down injured in the second half; Eric Bailly had to be replaced by Matteo Darmian, leaving Jose Mourinho's side down to the bare bones at centre-back.
Bleacher Report's Nick Akerman suggested this injury is likely to be a crucial one for United:
Swansea tried to capitalise on these defensive shifts by throwing men forward, and there were signs that the Red Devils back four was beginning to creak. Sigurdsson fired over after some fluid interchanges from the Welsh club.
The Iceland international merely seemed to be finding his range, though. And when Swansea were awarded a free-kick 11 minutes from time, Sigurdsson guided a pinpoint effort over the wall, with De Gea rooted to the spot. Per Squawka Football, the midfielder is so dangerous from set pieces:
It left the Red Devils with around 10 minutes to respond, and the home side started to press forward, while the visitors were content with a point. However, Swansea were looking a threat on the counter as the game became stretched, and Llorente missed a golden chance at the death to secure an unlikely win for the visitors.
Eventually, neither side was able to find what would have been a crucial winner, although the celebrations at the final whistle from the Swansea supporters suggested they were delighted with the point. The Red Devils will surely rue this missed opportunity in their chase for UEFA Champions League football.