Ronny Deila's men came into Sunday's clash unbeaten in their last eight league matches, but goals from Nadir Ciftci and Stuart Armstrong secured all three points for United, who move into second place.
Although Celtic dominated in terms of possession and chances, it was Jackie McNamara's side who showed a clinical touch in front of goal, with Ciftci influential throughout
The 22-year-old took advantage of a mistake by Efe Ambrose after five minutes to open the scoring, the striker calmly steering his eighth goal of the season past Craig Gordon.
Celtic then laid siege on United's goal with Stefan Johansen denied on two occasions, first by Radoslaw Cierzniak and then by the post.
As half-time approached the equalising goal seemed inevitable, but it never came, with Virgil van Dijk going closest as his header was cleared off the line.
The woodwork denied Celtic again early in the second half, and five minutes after the hour some slack defending enabled United to double their advantage.
Ciftci was again involved as he beat two men inside the penalty area before lifting a cross to the back post where the unmarked Armstrong had the simple task of heading home.
Celtic introduced Kris Commons, Leigh Griffiths and Stefan Scepovic to try to find a way back into the game, and the latter thought he had done so with 10 minutes to go only for the offside flag to deny him a second goal of the campaign.
Griffiths did pull a goal back with three minutes to play courtesy of a composed finish, and he could have rescued a point in stoppage time but the woodwork again came to the hosts' rescue.
