It was a game in which the Terrors had dominated but looked like they had been beaten by the brilliance of Shunsuke Nakamura.
After their hammering of Hearts, the Tangerines continued where they had left off creating the first chance in the third minute with a left-foot volley from Daly which was easily gathered by Artur Boruc.
And six minutes later Mark Kerr came within a hairsbreadth of opening the scoring. Latching on to a poor defensive clearance, the midfielder struck a 25-yard shot past the outstretched arms of the keeper.
The visitors eventually conjured the first chance in the 12th minute with Gary Caldwell heading Aiden McGeady's corner narrowly past from 12 yards.
Seconds later Nakamura's cross was brilliantly cleared by David McCracken as Craig Beattie looked set to pounce.
A 22nd minute blast from Barry Robson had Boruc in trouble as the keeper spilled the ball only to be saved Darren O'Dea who booted the ball to safety.
But Celtic had to thank their keeper on the half-hour when Steve Robb cut inside Lee Naylor and cracked a curling 20-yard shot which, with the lightest of fingertip touches, flew off the face of the bar and narrowly out of reach of the on-rushing Daly.
The Hoops' frustration was evident in the 33rd minute when Neil Lennon's through ball saw Beattie fly in with the studs into United keeper Derek Stillie and receive a caution for his misdemeanour.
Celtic were struggling but their best player of the half, McGeady, tried to lift them in the 37th minute with a wonderful 40-yard run and shot which Stillie had to push away.
Against the run of play, the Hoops took the lead with a brilliant goal from Nakamura.
Following a soft foul by McCracken on Beattie, the Japanese midfielder struck a beautiful 22-yard curling shot over the wall just inside the post leaving Stillie stranded.
But the Tangerines kept pushing and another Robson shot in the 57th minute slid narrowly past with Brouc beaten.
An excellent 73rd minute move from the Celts almost produced a second with Jan Vennegoor of Hesselink pulling the ball across goal and just past the post.
Just before United scored they went down to ten men three minutes from time. Naylor had a kick at Robson and while the Celt was booked, the United captain squared up to his opponent and earned his marching orders.
But then came the stunner for Celtic with a fully-deserved equaliser for the Terrors.
Stuart Duff's long ball was chested down by Daly and the big striker volleyed home from 18 yards.