In a dull first-half Manchester United stole the lead in stoppage-time via a Wayne Rooney penalty.
The second-half saw Swansea display the resilience of a side fighting for survival and were duly rewarded with a timely equailiser from captain Gylfi Sigurdsson in the 79th minute - to earn a crucial point.
Jose Mourinho's injury-hit squad knew three points would be enough to see them up to third place in the English Premier League, ahead of Liverpool and Manchester City.
However, fatigue and a lack of dynamism in midfield made for painful watching for the Old Trafford faithful in the opening stages of the encounter. Without the creativity of Paul Pogba and Juan Mata, the Red Devils often looked short of ideas in the final third.
The Swans midfield trio of Ki Sung-yueng, Tom Carrol and veteran Leon Britton controlled proceedings as the home side struggled to gain possession, but going into the break, they somehow found themselves a goal behind.
Lukasz Fabianski was adjudged to have brought down forward Marcus Rashford in the area and captain Rooney slotted home the subsequent spot-kick to hand Mourinho's side a controversial lead just before half-time.
Paul Clement's men could have felt hard done by coming out into the second 45 minutes, but they took the disappointment in their stride and kept toiling away at the Premier League giants without neglecting their defensive duties.
As the half wore on and clear cut chances were limited, the injuries started piling up for both sides once again. Eric Bailley had to make way with Matteo Darmian replacing the Ivorian at centre-back, while Jefferson Montero spent only five minutes on the pitch before he had to be stretched off the field.
However, the visitors were soon rewarded for the industrious display, with captain Sigurdsson netting a spectacular equaliser.
The Iceland international playmaker found the top corner of David de Gea's net, with the Spain international keeper rooted to his spot, a mere spectator to the fabulous strike.
Clement's side held on to the point with 10 minutes to go, to pull two points closer to 18th place Hull City - while United missed the chance to move into the Champions League qualification spot, with their tenth league draw at home this season.