Rooney swept home from the spot moments before half-time after Marcus Rashford had gone to ground under almost no contract from Lukasz Fabianski, but Swansea's set-piece maestro stole a point that could wreck the hosts' hopes of qualifying for the Champions League through domestic competition.
Swansea, who are now just two points adrift of safety in 18th place, looked the brighter of the teams for much of the first half but chances were limited before the penalty incident in stoppage time.
Jose Mourinho was without injured trio Paul Pogba, Chris Smalling and Phil Jones and Shaw and Bailly were forced off in either half to compound their concerns ahead of a crucial run of fixtures that includes trips to Arsenal and Tottenham alongside that European semi-final.
United looked likely to hold onto the three points regardless but Sigurdsson's stunning free-kick snatched a draw for the visitors to deny Mourinho's men the chance to move into the top four.
3 - Sigurdsson has become the first player to score in 3 consecutive PL away games at Old Trafford since Aguero in April 2015. Precision.— OptaJoe (@OptaJoe) April 30, 2017
A sluggish start from United was made worse when Shaw hobbled off with less than nine minutes played, the full-back indicating discomfort in his left leg.
Swansea were by far the better team in the opening quarter of an hour and Fernando Llorente brought a fine reaction stop out of David de Gea from a Tom Carroll cross, before United's shaky defence scrambled the ball clear.
First-half goals have been in short supply at Old Trafford this season but Jesse Lingard almost the broke the deadlock in the 17th minute, racing onto Anthony Martial's delivery before seeing his volley expertly blocked by Fabianski.
The Swansea keeper saved well again when Martial cut in from the left and drilled low at goal, but De Gea went one better when he stuck out a foot to stop Jordan Ayew after some fine footwork in the area.
The visitors' good work was undone in controversial fashion on the stroke of half-time, however. Rashford went down as Fabianski raced out to meet him and referee Neil Swarbrick pointed to the spot, after lengthy consideration.
Replays showed the England youngster had gone to ground far too easily and Rooney compounded Swansea's frustrations as he swept the penalty into the bottom-right corner.
Rooney thumped a clear chance against Martial's back three minutes after the restart as United pushed for a second, but there was more bad news for Mourinho as Bailly had to be replaced with what looked to be an injury to his right ankle.
Sigurdsson wasted a great chance to level on the break, blasting high from the edge of the area, but the Iceland midfielder responded in style with 11 minutes to play, bending a sublime free-kick past the motionless De Gea from 20 yards out.
It could have been worse for United had Llorente managed to connect with a Sigurdsson delivery with the goal gaping, but the teams had to settle for a draw that was greeted with huge frustration by the home fans who stayed until the final whistle.
Key Opta stats:
- Manchester United have set a new club record for the longest unbeaten run within a single top-flight season (25 games, W13 D12).
- Only Alan Shearer (190) has scored in more different Premier League games than Wayne Rooney (150).
- Gylfi Sigurdsson has been directly involved in 21 of Swansea's 40 Premier League goals this season (nine goals, 12 assists).