The Hoops were leading through goals from Miku, his first for the club since signing on loan from Getafe in the summer, and Tony Watt and looked certain to take the points back to Glasgow.
However, in the final minute of normal time United substitute Gary Mackay-Steven, on for Richie Ryan, scored what looked like a late consolation only for Efe Ambrose, on for the injured Thomas Rogne, to put a Barry Douglas cross past his own goalkeeper Fraser Forster in injury time.
Celtic went back top of the SPL table but it was not the end to the game manager Neil Lennon was looking for ahead of Wednesday night's Champions League clash with Barcelona at Celtic Park.
However, United boss Peter Houston, who watched his side lose against Hearts on penalties in their Scottish League Cup quarter-final in midweek, will be delighted at taking a point from a game that looked all over when Watt scored with 10 minutes remaining.
Lennon made three changes to the team which beat St Johnstone 5-0 in the League Cup on Tuesday night.
Adam Matthews, Rogne and Charlie Mulgrew came in at the expense of Mikael Lustig, Ambrose and Gary Hooper, the latter not even on the bench.
Houston brought in Douglas for Michael Gardyne, who started as substitute, which allowed skipper Jon Daly to resume his place in attack.
It was not until the 13th minute that fans witnessed their first real piece of goalmouth action when Rogne headed Scott Brown's cross from the right past the post from no more than eight yards.
The visitors were busier and more purposeful but suffered a blow in the 20th minute when Ambrose had to replace Rogne after the Norwegian stopper limped off with a leg injury.
Moments later, Celtic threatened again when midfielder Victor Wanyama burst powerfully through the middle and when Kris Commons sent a cross from wide on the right to the far post, Watt headed over.
It took a timely intervention by Douglas in the 28th minute to prevent Commons getting a clear strike on target from 14 yards after the Hoops midfielder had been played in Brown, with the ball trickling back to Radoslaw Cierzniak.
The home side were very slowly making their way into the game and in the 35th minute United midfielder Stuart Armstrong finished off a fine move by thrashing a drive from 25 yards over the bar.
Celtic reasserted themselves before the break and should have taken the lead five minutes from the interval.
Mulgrew's whipped-in cross from the left found Brown unmarked inside the box but the Celtic skipper failed to get a proper connection on the ball with his head and it flew wide.
The champions almost took the lead less than a minute after the restart when Watt released overlapping left-back Emilio Izaguirre inside the United box but the Honduran hit the outside of the post with his left-footed effort.
Almost immediately Izaguirre was replaced by Miku, who went up alongside Watt, with Mulgrew moving to left-back.
Moments later United missed their best chance of the match. Former Celtic midfielder Willo Flood's powerful long-range shot was spilled by Forster and Johnny Russell blazed over the bar from around eight yards out.
Then, after Wanyama was booked for a foul on John Rankin, from Douglas's free-kick United defender Gavin Gunning stabbed the ball wide from six yards.
But when Celtic's
landmark opener from Miku arrived soon afterwards it was well deserved.
Wanyama combined with Watt, who played in the Venezuelan, and he calmly chipped the ball over Cierzniak from 10 yards and raced to take the acclaim of the Celtic fans, before picking up a booking by referee Willie Collum for his troubles.
The home side had only themselves to blame when Watt grabbed the Hoops' second with 10 minutes remaining.
A poor clearance from Cierzniak arrived at the feet of the teenage striker and he bore down on goal before rifling in a shot before he too was booked for leaving the pitch.
It looked to be all over for United, even when Mackay-Steven guided the ball past Forster after Kelvin Wilson had cleared a shot from United substitute Rudi Skacel, on for Armstrong.
However, there was more drama to follow in the first minute of added time when Ambrose, at full stretch, headed a Douglas cross into his own net to leave the travelling fans and Lennon bemused.