Despite their lack of chances in front of goal, the Tangerines' solid defensive performance, in the face of almost constant pressure from the Celtic strikers, will have boosted their confidence ahead of Sunday's CIS Cup final against the other half of the Old Firm.
Shunsuke Nakamura and Scott McDonald both presented a danger for the United defence early on.
The Japanese midfielder cracked a volley off the post early on after good work from Aiden McGeady.
Minutes later Nakamura and McDonald linked up well on the break only to see the Australian's effort blocked.
Nakamura was the provider again, this time for Georgios Samaras who headed his effort narrowly over the bar and Scott Brown got in on the act scuffing his shot just high of the target.
It wasn't all one-way traffic as United battled to trouble Artur Boruc at the other end. Danny Grainger saw his shot deflected around the post for a corner kick.
But the visitors could not sustain the pressure and Celtic broke back as Brown had a header turned around the post by Lukasz Zaluska and McDonald hit the woodwork.
On 38 minutes Craig Conway passed up the best scoring chance of the match for the Tangerines. Michael Kovacevic made a great run from midfield, holding off McGeady before producing a pinpoint cross for Conway who headed over the crossbar.
Celtic ramped up the pressure on their opponents at the start of the second half and the visitors struggled for the first few minutes to even clear the ball from their own half for any period of time.
Brown tested the defence again with a glancing header that skimmed across the face of goal while Nakamura narrowly missed the top corner of the net with a free-kick from just outside the area.
Barry Robson was brought into the battle midway through the second half adding an extra edge to the game as he lined up against his former team-mates.
And there was no let up on the Dundee United goal as Zaluska produced two excellent saves, first from Stephen McManus and then bravely at the feet of Gary Caldwell.
It was the start of a frantic period for the visiting goalkeeper who then had to save from Nakamura before blocking a follow up from substitute Jan Vennegoor of Hesselink.