The Tannadice side looked to be heading for their first win in the east end of Glasgow for 21 years as they resolutely defended a lead given to them in the 38th minute by midfielder Stuart Armstrong.
That long-awaited win would have been almost certain had striker Brian Graham not missed an incredible sitter in the 65th minute when he somehow made a hash of a Nadir Ciftci cross, and the visitors were made to pay in the second minute of three added on as Mulgrew rose to flick a cross from Emilio Izaguirre into the net.
Celtic had been profligate all afternoon but the equaliser kept them unbeaten in the league after 11 games as they look ahead to their crunch Champions League clash with Ajax in Amsterdam on Wednesday night, where victory could strengthen hopes of qualifying through to the knockout stages.
Amido Balde was given a start for Celtic after scoring the winner at Partick Thistle last week.
The 22-year-old Portuguese striker was one of four changes with Darnell Fisher, Charlie Mulgrew and James Forrest also in and Georgios Samaras, Beram Kayal, Teemu Pukki and Mikael Lustig out, the latter three starting on the bench.
Ciftci, who has been accused of assaulting an assistant referee by the Scottish Football Association after being sent off in the Scottish League Cup defeat at Inverness in midweek, kept his place as Brian Graham started as lone striker.
Celtic Park was sodden by kick-off, which lent itself to a plethora of chances created and missed before Armstrong's goal.
Anthony Stokes, who signed a three-year extension to his contract in midweek, missed a couple of opportunities within the first five minutes, volleying over the bar from the edge of the box and then heading a Mulgrew free-kick over the bar.
The Republic of Ireland international, Balde and Mulgrew all had efforts while Hoops keeper Fraser Forster had to make decent saves from Gary Mackay-Steven and Graham.
A goal-bound Efe Ambrose header from a Stokes corner in the 35th minute was cleared by United left-back Andrew Robertson before Tannadice stopper John Souttar headed Mark Wilson's free-kick past the post.
It seemed like a goal would never come but three minutes later it did and it was worth the wait.
When Graham knocked the ball back to Ciftci 25 yards from goal, Armstrong timed his run perfectly to latch on to a delightful chip and rounded Forster before knocking it into the net for his fourth goal of the season.
Celtic fought back immediately and Forrest's drive, after Cierzniak had spilled a corner, was blocked by a wall of orange jerseys for another corner which came to nothing.
Stokes had two more unsuccessful efforts before the break, flashing a shot over the bar from six yards before having a low drive saved by the legs of Cierzniak.
Lustig replaced Fisher for the start of the second half and, as expected, the home side pressed for the leveller but the profligacy continued.
Balde headed a Mulgrew corner over the bar in the 55th minute and after a disappointing afternoon he was replaced by Pukki just after the hour mark as Lennon tried another avenue.
As the crowd still wondered how Graham had managed to get Ciftci's cross tangled between his legs, Pukki sent a drive whistling past the post.
Dutch winger Derk Boerrigter, back after a long spell out with an ankle injury, replaced Joe Ledley in the 72nd minute but with nine minutes remaining it was Stokes who tested Cierzniak again at his near post but again the United keeper proved up to the task with a fine save.
There was a desperation about Celtic in the final stages but as so often happens at Parkhead, there was to be a twist in the tale, coming when Mulgrew got his head on Izaguirre's cross.
- Sports & Recreation
- Brian Graham
- Nadir Ciftci