Gordon Strachan's men started the match well and Shunsuke Nakamura went close early on for the reigning SPL Champions. Lee Naylor swung in a terrific ball from the left and Nakamura ghosted in before firing wide of target from 12 yards.
But it was Dundee United who took the lead against the run of play. The Celtic defence failed to deal with a harmless long ball forward from Lee Mair and David Robertson was the first to react, heading the ball past Artur Boruc.
Having set up Robertson's goal, Mair was then alert at the other end to deny the Hoops an equaliser. Jiri Jarosik sent in a cross and Maciej Zurawski looked certain to score, but his effort was superbly blocked by Mair.
The Celtic defence was looking decidedly shaky at times and the porous Parkhead goal was breached for a second time in the match after 57 minutes.
Once again the Celtic back four failed to deal with a high ball into the box and Collin Samuel was on hand to net from close range.
Celtic stepped up the pace as they sought to get back in the game and second-half subs Kenny Miller and Derek Riordan were both denied by fine blocks from Derek Stillie.
But Celtic were not to be denied and two goals in ninety seconds saw the Hoops draw level.
Darren O'Dea headed home to pull one back before Nakamura fired home a sensational equaliser.