The home team had Scott McDonald to thank for a well-taken hat-trick as the striker begins to look more and more like an excellent purchase by the Bhoys.
United had their chances, especially when Celtic were only one ahead, but appeared to lack belief that they could take anything from this match.
Celtic made an early breakthrough when Aiden McGeady easily cut past a static Barry Robson before crossing to the back post where Jan Vennegoor of Hesselink's header was desperately clawed out by the United keeper, only for McDonald to force the ball home from point-blank range.
United were looking very unsure at this stage and constantly gave away possession.
If McDonald's pass had carried just a little more weight, it would have put Vennegoor of Hesselink clear on goal.
A swift break by Massimo Donati led to McDonald laying the ball back to the Italian midfielder, but his weak chip went straight into the arms of Grzegorz Szamotulski.
Vennegoor of Hesslink limped off after a challenge with Darren Dods before McGeady's swerving shot was pushed away to safety by the keeper.
Shortly after the interval Celtic should have added to their lead. Jean-Joel Perrier Doumbe's cutback was touched on by McDonald to Chris Killen, but from eight yards the striker placed his shot wide.
It was a bad miss and Celtic nearly paid the penalty when the Terrors should have equalised. Jordan Robertson beat the offside trap, but Artur Boruc blocked brilliantly with his legs to deny the youngster.
Robson's inswinging corner then found Dodds unmarked at the back post but the centre-back headed wide from six yards.
United were made to pay as a powerful run by Scott Brown into the box led to Celtic's second goal.
Brown's pass reached McDonald who rounded the keeper before scoring from an acute angle.
The match was over for the visitors when they fell three behind.
McGeady, who tormented the United defence all afternoon, skipped past Stuart Duff before delivering a great cross to the back post which McDonald volleyed home for his hat-trick.
McDonald went off to a standing ovation and this victory will put Celtic in good heart for the visit of AC Milan On Wednesday.