In fact, in a sparkling first half, United set their stall out by going ahead with only three minutes on the clock.
The Tangerines took a shock lead with an own goal from Chris Sutton as Barry Robson's curling corner flew off the big striker, giving his goalkeeper no chance.
Shaun Maloney tried to haul the Bhoys back into it the 10th minute, falling to the ground under Paul Ritchie's challenge, but the referee rightly ignored his theatrical efforts to win a penalty.
Then Sutton attempted to redeem himself with a glancing header which went narrowly wide, but the miss only delayed the inevitable leveller.
John Hartson latched onto a 17th-minute dummy from Sutton and slid the ball home from 10 yards after good wing work from Paul Telfer.
This inspired Celtic and after some exciting, incisive midfield play they were unlucky not to extend there lead until the 28th minute, when Maloney's shot came off the legs of Sutton and deflected into the net for a fortuitous goal.
But the visitors were stunned two minutes later when Collin Samuel scored a wonderful equaliser, taking the ball on the turn from Mark Wilson and drilling home a low shot from 16 yards.
With both defences looking uneasy further goals were inevitable and Celtic's third arrived two minutes later. Alan Archibald, under pressure from Sutton, deflected the ball past his own keeper with his shoulder.
In a quieter second half, Hartson should have extended Celtic's lead in the 63rd minute when he dived to head Telfer's free-kick, but Derek Stillie produced a magnificent diving save to deny the Welshman.
Gordon Strachan's men secured the points two minutes from time when substitute Stephen Pearson slid in to convert Stilian Petrov's inviting, low cross.