With a view to Wednesday night's Champions League clash with Ajax in Amsterdam, the Hoops boss left Beram Kayal, Mikael Lustig and Teemu Pukki on the bench, while Georgios Samaras (knee) and Kris Commons (hamstring), almost certain to start in the crunch Group H game, had an afternoon off.
The former Celtic captain then watched as his side huffed and puffed for most of the afternoon.
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.
"I'm very happy that we got something we deserved," said Lennon.
"I should take the criticism if there's any coming our way because I picked the team.
"I wanted to use the squad for this game, with a huge game on Wednesday.
"I made four or five changes, which isn't ideal, but players need game time.
"I know some of them are a little short because of the injury problems we've had but Lustig will come through okay, Kayal will be fit, Georgios Samaras and Kris Commons will be fit, so we'll be a lot stronger on Wednesday.
"So I'll take a point. It preserves the unbeaten record which obviously we're very proud of but we were a bit flat and certainly in the final third we need to do better.
"We had umpteen chances, we had three off the line in the first half and so much dominance in the second half. And we were sloppy.
"That comes down to my team selection but we made the changes when we had to and got the goal we deserved.
"We've had a heavy, heavy load of games, but I'm not using that as an excuse.
"The only excuse or criticism is my team selection. Don't blame the players today."
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.
Visiting manager Jackie McNamara admitted the result felt like a defeat but refused to be too harsh on his players.
"I think Brian (Graham) was in two minds how to take it, although that possibly could have killed the game for us," said the former Celtic defender.
"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 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 (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."