Sam Vokes scored a third for Burnley but it was ruled out for offside, before Lingard struck at the death to earn United a point and avoid defeat.
Alvaro Morata scored on his return to the Chelsea team following suspension as Brighton were beaten 2-0.
A half-time substitute - coming on for the disappointing Zlatan Ibrahimović - Lingard arrived on the scene with his side 0-2 down.
United will still be without a string of first-team players as they look to get back to winning ways. United went into this latest round of fixtures having only conceded three goals at home, and two of those were at the hands of Pep Guardiola's rampant City, which put them level with Liverpool in terms of which side had the best defensive record at home. There are many big clubs and you say big clubs, I know what is a big club. "The end product of so much dominance, two goals, is a miserable number".
City, who are on 55 points, travel to Newcastle United tomorrow.
"Sometimes you make mistakes and you are not punished, but we were punished".
Sean Dyche's men had been well on course for another dream result in a season to remember, leaving them in the hunt for a top-four finish, but they could not hold off their opponents and eventually came undone late on.
Asked if they are in a rough patch, he replied: "Why?"
You suspect that Rashford was one of the players Mourinho had in mind when he blasted his players during his post-match interview with Sky Sports. I talked to the players at half time when it was 1-1 that it was an easy match to win.
"It's the first Christmas without my father and I think that's the most hard one", he said. "The game plan is you're going to have to defend and do all of the ugly details of the game well". We made some fantastic opportunities to score, just after this, we conceded the free-kick.
"But where we were a year ago to where we are now, I'm very, very pleased and that shows a lot of development in the team and a lot of development in my staff and what we're trying to achieve".
"Yes, and I had the chance to speak with Ashley Young to tell him the way the team had to be in position because of Smalling's difficulty, but I think a lack of maturity because I can not stop the game to give a team-talk for the last two minutes".