The late substitute's goal also meant Rangers stretched their lead at the top of the Scottish Premier Division to seven points with 11 games left, despite being held 0-0 by Hearts at Ibrox.
Nevertheless, Celtic's manager, Tommy Burns, was in no mood to surrender. "There will be a few more twists and turns in this race I can assure you and we will dust ourselves down and start again on Monday," he said.
Celtic, already without Andreas Thom, Simon Donnelly, Stuart Gray, Brian O'Neil and Morten Wieghorst, also had Tom Boyd ruled out through injury, so their record signing Alan Stubbs returned for his first game since the 2 January defeat by Rangers.
United set the pace with central defender Mark Perry miscuing when well placed and their veteran midfielder Jim McInally shooting tamely when found inside the box by Kjell Olafsson.
Celtic's Pierre Van Hooijdonk might have done better than hit the side- net when he was presented with an angled shooting chance inside the box, and United's goalkeeper Sieb Dykstra made a superb one-handed stop in the first minute of the second-half as Celtic finally came to life.
Olafsson and Robbie Winters both went close for United but Celtic could have stolen it when their Portuguese top scorer Jorge Cadete twice came within an ace of scoring. First he picked up a pass by Paul McStay and rounded Dykstra only for Perry to clear off the line. Then he met a cross from Van Hooijdonk but his header struck the bar.
McSwegan came on in the 83rd minute for Winters and finally broke the deadlock with his eighth goal of the season. Released by Olafsson, he tricked his way past Jackie McNamara and then coolly lobbed his shot over Stewart Kerr.Reuse content