A first-half header from defender Glenn Loovens and a deflected Georgios Samaras strike seven minutes after the break had appeared to take the game beyond the Tannadice men, who had, at times, out-passed the champions.
However, Craig Levein’s side showed commendable spirit and David Robertson’s strike after 57 minutes ensured a tension-filled end to the game.
But the champions hung on, doggedly at times, and are now a point ahead of rivals Rangers – who play Hibernian at Easter Road tonight.
Goalscorer Loovens acknowledged the Glasgow side had made life difficult for themselves after letting the opposition back into the game at 2-0.
“We are happy with the three points,” Loovens added. “At times we played some great football but at 2-0 it should be over – but we made it too hard for ourselves in the end.
“We knew we had to be strong in the end to defend the three points.”
Loovens admitted Celtic are also dependent on their city rivals to drop points.
“We are depending on them to lose points and we have to make sure we win our next two games. The pressure is on them now.”
Aiden McGeady returned to the Celtic line-up after being controversially dropped for the Old Firm defeat at Ibrox on Saturday, and that was the pre-match talking point.
Loovens’ 21st minute opener arrived after Shunsuke Nakamura flighted a corner into a busy six-yard box and the big Dutchman rose highest to flick the ball past Zaluska.
The game seemed over in the 52nd minute when Samaras made it 2-0 with a fortuitous goal.
Brown combined with McDonald to set up the Greece international 20 yards from goal. His shot initially appeared harmless, until it took a clear deflection off Wilkie to wrong-foot Zaluska.
In the 57th minute, Robertson reduced the lead when he appeared at the far post to dispatch Paul Dixon’s deep cross past Boruc.
After some nervy final moments, referee Stuart Dougal’s full-time whistle was greeted with relief from the Celtic fans.
CELTIC: Boruc, Hinkel, Caldwell, Loovens, Naylor, Nakamura (Flood 85), Scott Brown, Crosas (Hartley 61), McGeady, McDonald, Samaras (Vennegoor of Hesselink 76).
DUNDEE UTD: Zaluska, Kovacevic (Dillon 88), Kenneth, Wilkie, Dixon, Swanson (Shala 58), David Robertson, Buaben, Gomis, Conway, Sandaza (Goodwillie 81).
Referee: S Dougal.
SPL result: Aberdeen 0 Hearts 0