The stadium was left stunned in the first half as the Clarets claimed a 2-0 lead via Ashley Barnes' close-range finish in the third minute and Steven Defour's superb strike in the 36th.
Having come on for Zlatan Ibrahimovic at the interval, Lingard then pulled a goal back eight minutes after the break with a back-heel before striking an equaliser in stoppage time.
United boss Jose Mourinho stated afterwards: "The problem with the first-half performance is another 's' goal - you are intelligent and don't need me to tell the whole world the other letters in the word. What we are having now is lots of really very bad goals against us, so that obviously has an impact in the game.
"At 2-0, it makes things more difficult at half-time, I need my players to believe in them[selves] and I need them to believe in me; to accept the risk and not be afraid to lose three or four. To not be afraid of what was possible - what didn't happen but it could happen, that there is a negative reaction from the supporters."