Ashley Barnes and Chris Wood had put Burnley on course for a first win at Old Trafford since 1962 but Paul Pogba and Victor Lindelof rescued a point for the hosts, who had been frustrated for most of the game by an organized side backed up by in-form goalkeeper Tom Heaton.
Solskjaer was looking for a ninth straight win in all competitions — and a record-breaking seventh at the start of a Premier League reign — but United missed the chance to move level on points with fourth-place Chelsea in the race for Champions League qualification.
"The comeback was brilliant, the players have been asked about how they would react if they went 1-0 or 2-0 down so I felt the comeback was fantastic," Solskjaer said. "We had loads of chances, loads of possession, we didn't deserve to lose. But we did it to ourselves really.
"We fought, that wasn't the problem. We were just too slow in possession in the first half ... We disappointed ourselves and we disappointed the crowd today, but the ones who stayed had a typical Old Trafford finish to the game so that was a big plus."
Burnley moved four points clear of the relegation zone. "It's a tough one to swallow," former United goalkeeper Heaton said. "We were running on empty at the end. Obviously it's a little sore right now but going forward it's an excellent point for us. They have been on a great run but we just focused on ourselves, kept our shape right, restricted them, had no fear and I think we approached it well ... We feel like we have turned the corner a little but it's important we take that momentum forward."
Barnes put the visitors ahead in the 51st minute after Andreas Pereira was caught in possession by Jack Cork, and Wood appeared to put the game out of reach when he was afforded space to send a fine cross by Ashley Westwood past David De Gea in the 81st.
But Pogba fired home a penalty with three minutes remaining after Jeff Hendrick had hauled back Jesse Lingard and Lindelof equalized in stoppage time in a comeback the watching Alex Ferguson would have been proud of.
had denied Pogba and then Alexis Sanchez, but Lindelof reacted quickest to the rebound and directed home to preserve Solskjaer's unbeaten run. It is the second successive season that United has secured a 2-2 draw against Burnley at Old Trafford in stoppage time.
More AP English soccer: https://apnews.com/PremierLeague and https://twitter.com/AP_Sports