The international midfielder, whose two years at Parkhead have been blighted by a series of injuries, supplied the winning goal on his return after overcoming a thigh problem which has kept him out since the opening day of the season.
For a team whose spirit has been sapped by three costly League games without success, and rendered inactive by postponements in the last fortnight, O'Donnell's contribution was very welcome.
Celtic were without their top scorer Pierre van Hooijdonk - the official line being that the Dutchman was still suffering from the hamstring injury he apparently picked up while sitting on the bench for his country on World Cup duty last weekend. However, it seems likely that the bitter impasse over a wage increase will hasten his departure, with Feyenoord ready to offer pounds 2m for the player.
United, with just one defeat in 11 games, were in buoyant mood, and Robbie Winters almost pumped that optimism up another notch in the fifth minute when he latched onto a pass from Lars Zetterlund, but the young striker's finish was just a yard wide of the post.
Another of United's trio of Scandinavian imports, Kjell Olofsson, came closer in the 17th minute with a shot from the edge of the area which forced the Celtic keeper, Stewart Kerr, to make a fine save to his right.Eight minutes later, Olofsson again got clear from the Celtic defence but Kerr once more proved equal to the situation.
Van Hooijdonk's stand-in, Chris Hay, then cut a swathe through a crop of orange shirts before delivering a telling cross, but O'Donnell stumbled just as he looked set to apply the finishing touch from six yards.
United could have taken the lead in the 38th minute when Winters produced an inventive hook-shot with his back to goal but again Kerr's alertness and agility kept Celtic alive.
The timing of the save could not have been better, for within 60 seconds the home side had scored. Andreas Thom engineered the opening with a cross for Simon Donnelly, who returned the ball to the back post where this time O'Donnell made no mistake with a powerful header.
Celtic thought they had gained the opportunity to stretch their lead in the 49th minute when Jorge Cadete appeared to be brought down in the area by Erik Pedersen, but, to the fury of the Parkhead crowd, the referee dismissed the claim.
The Portuguese forward twice combined with Thom to threaten the United goal, but on both occasions the final pass ruined their good work.
O'Donnell marked his return in a less auspicious fashion in the 57th minute when he became the game's first booking, for pulling Winters' shirt as the Dundee United player outpaced him.
Celtic's search for a second goal deserved better than the fate which met some sublime skill by Thom in the 63rd minute. The German got a deft touch on Tosh McKinlay's free-kick but saw his header strike the bar.Reuse content