Celtic went two ahead through goals by Georgios Samaras, but a marvellous free-kick by Paul Dixon and the crisp finish by substitute Warren Feeney enabled the visitors to draw level.
This draw allowed Celtic to move eight points clear at the top of the table, but when they went two up the home team must have thought the match was over.
Celtic started brightly and Scott Brown and Samaras both went close before the Greece striker opened the scoring.
Brown chased down a long ball before dispossessing Lee Wilkie and squaring the ball to Samaras, who drilled his shot home from 18 yards.
United should have equalised when Wilkie slid a pass through to Scott Robertson but he curled his shot wide from 10 yards.
Shunsuke Nakamura cut in from the right to test the United keeper Lukasz Zaluska, before Barry Robson saw his shot deflected for a corner.
After the break, Celtic went two up when Nakamura's free-kick after Scott McDonald was fouled was headed home from six yards by Samaras.
Stephen McManus was booked for a bad foul on Morgaro Gomis at the edge of the box and Celtic were punished when Dixon curled his free-kick over the wall into the top corner.
At the other end, Dixon cleared Brown's shot off the line after Zaluska had flapped McDonald's cross.
But it was the visitors, with Gomis everywhere, who looked the more threatening.
Feeney intercepted the wayward pass back from Robson but, after rounding Boruc, his shot lacked power and McManus was able to clear.
Feeney made amends when he latched on to Jon Daly's flick to toe-poke the ball past Boruc and earn United a deserved draw.