United looked to be heading for their first win in the east end of Glasgow for 21 years as they defended resolutely a lead given to them in the 38th minute by midfielder Stuart Armstrong.
However, after passing up a series of chances, the home side eventually levelled in the second of three added minutes when Mulgrew rose to head a cross from Emilio Izaguirre past Radoslaw Cierzniak to make it 11 league games unbeaten for the league leaders.
United's long-awaited win at Celtic Park would have been almost certain had striker Brian Graham not missed a sitter in the 65th minute when he somehow made a hash of a Nadir Ciftci cross, and had goalkeeper Cierzniak not hastily got rid of the ball in the dying seconds before Celtic regained possession and levelled.
McNamara, a former Celtic defender, said: "I think Brian (Graham) was in two minds how to take it, although that possibly could have killed the game for us.
"He mentioned it when he came off, it was obviously playing on his mind but to be honest I thought he was excellent. His play deserved a goal.
"I would never criticise the keeper, it is part of learning and he realises it himself.
"He had a fantastic save from [Anthony] Stokes five minutes before that.
"It shows how far we have come when we are disappointed to go away with a point.
"But we need to learn to see it through. In injury-time our 19-year-old left-back (Andrew Robertson) has nearly got down the right-hand side.
"It is enthusiasm, I wouldn't try to take that side of our game out, but sometimes you just see it through and be professional."