Kris Commons and Anthony Stokes scored twice to lead the reigning champions to an easy win that erased a few of the bad memories of their Champions League defeat by Juventus in midweek.
The home side were soon on the back foot and they fell a goal behind in the 10th minute when Gary Mackay-Steven stole the ball from Charlie Mulgrew in midfield and ran into the box before feeding an unmarked Stuart Armstrong for an easy finish.
But the lead lasted less than two minutes as Celtic responded from a corner at the other end, Efe Ambrose rising to head Commons' centre past Radoslaw Cierzniak.
The early turnaround was completed on 22 when Commons got his name on the score-sheet, rifling Emilio Izaguirre's cross into the net off the crossbar.
United were given a chance to level four minutes later when Mackay-Steven was harshly adjudged to have been brought down in the box by Adam Matthews, but Fraser Forster made a fine save to deny Jon Daly from 12 yards.
That escape seemed to inspire Celtic and they took a 3-1 lead eight minutes before half-time thanks to persistent work from Stokes, who set up Joe Ledley for a curled shot into the top corner of Cierzniak's net with the aid of a deflection.
The game was over as a contest 10 minutes into the second-half when Celtic made no mistake with a penalty of their own, Commons grabbing his second goal of the game after Gary Hooper had been bundled over in the area.
A fifth goal came 19 minutes from time when Stokes took a pass from Victor Wanyama and fired past Cierzniak from just outside the area.
Stokes scored his second, and Celtic's sixth, eight minutes from time as the rout continued, volleying past Cierzniak after James Forrest had been denied by a brilliant save from the Polish keeper.
There was still time for an eighth goal, although it was United who wrapped up the scoring in the final minute through the prolific Johnny Russell, but that was nothing more than a consolation on manager Jackie McNamara's return to the club he served so well as a player.