Kris Commons opened the scoring before Georgios Samaras bagged a first-half double with Anthony Stokes glossing the win five minutes from time.
Commons opened the scoring with just 11 minutes on the clock when he was brought down by Jon Daly and curled the resulting free-kick into the bottom corner.
Samaras doubled the lead six minutes later when his shot struck Jon Souttar and looped over a stranded Radoslaw Cierzniak and into the top corner.
The Greek striker had the ball in the net again on 20 minutes but was judged offside but he would not be denied a second when he headed in a Commons free-kick on 27.
The home side had not been without their chances in a lively first half and Fraser Forster had to be on his toes to deny Rory Boulding and Mark Millar in quick succession while Stuart Armstrong drilled wide after a neat one-two with Gary Mackay-Steven.
But Celtic looked to continue their dominance in the second half and Gary Hooper appeared to have added a fourth when he bundled in a Samaras cross only to be denied by the linesman's flag.
Joe Ledley brought a fine save from Cierzniak from a Commons corner before Foster produced another double save from Daly and Ryan Gauld.
The teenager was inches away from getting United on the scoresheet on 71 but could not quite get Brian McLean's cross under control.
It was the visitors who finished strongest, Hooper firing over after Samaras had a shot blocked and Scott Brown striking the foot of the post before Stokes wrapped it up with a deflected effort.
And it could have been even more convincing had debutant Bahrudin Atajic got his header on target but Neil Lennon will be more than happy with his side's warm-up for next weekend's Scottish Cup final against Hibernian.