Shaun Maloney and Chris Sutton's late goals stretched their winning league run to 23 games to match Morton's feat of 1963-64 and surely kill off Rangers' already slim title hopes.
Celtic now lead their Glasgow rivals at the top of the Bank of Scotland Premier League by 13 points.
Alan Archibald's 60th-minute opener for the visitors had threatened to end Celtic's magnificent league record and end an unbeaten run at Parkhead stretching back to 2001.
The Celtic defence were given an early warning in the second minute when the ball broke to Charlie Miller just outside the area, but he lashed a left-footed shot over the crossbar.
Celtic got their passing game together and Henrik Larsson attempted to find Stephen Pearson at the far post in the 10th minute, but Paul Gallacher coolly caught the ball.
A number of Tangerines players called for a penalty in the 17th minute when Jim McIntyre went down in the box following Mark Wilson's free-kick.
The home crowd started to become restless as Stilian Petrov's long-range effort was far too ambitious and Gallacher saved it easily.
Petrov was deservedly booked just before the break for a terrible tackle on Barry Robson to sum up Celtic's half.
Stanislav Varga had the perfect opportunity to put the home side ahead in the 52nd minute. Alan Thompson's free-kick spun up in the air off Sutton and from just six yards the defender headed wide.
United stunned the home crowd into silence on the hour when they snatched the lead. Robson crossed and Archibald stuck out a leg to direct the ball past Douglas.
Varga again should have scored just two minutes later when Alan Thompson's corner found its way to him, but it bounced off him and went straight at a relieved Gallacher.
However, the pressure told in the 80th minute when Maloney arrived at the perfect time to sidefoot Sutton's cross home.
The drama continued just a minute later when the home side were awarded a penalty following Stuart Duff's mistimed tackle on Petrov and Sutton sent Gallacher the wrong way.
Former Rangers man Billy Dodds was sent off in the dying seconds for two yellow cards to add insult to injury for a shell-shocked United.
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