Lingard, who had an excellent loan spell with the Hammers in the second half of last season, curled in an 89th minute winner.
But there was still time for more drama with West Ham awarded a 93rd minute penalty for a Luke Shaw handball.
Hammers boss David Moyes brought on Mark Noble for the spotkick but de Gea saved well one handed to his left.
Earlier, Said Benrahma’s deflected effort had put West Ham ahead on the half hour, before Cristiano Ronaldo poked in an equaliser five minutes later.
United boss Ole Gunnar Solskjaer said: We deserved to win. I think we created some big chances, we had pressure on them, should have had a few penalties, great response when they scored, good attitude, very good away performance in a difficult cauldron.”
The result moves United up to 13 points from five games but they are third, behind Chelsea and Liverpool who are joint top.
Chelsea kept up their strong start with a 3-0 win away to Tottenham with second-half goals from Thiago Silva, Ngolo Kante and Antonio Rudiger.
“We lacked energy and we spoke clearly about it at half-time but the second half was a very good performance and a well-deserved win,” Chelsea boss Thomas Tuchel said.
“It was a very good reaction so Im absolutely happy with our performance in second half.”
Elsewhere, Brighton & Hove Albion got past Leicester City 2-1, despite the Foxes rallying late, with two goals disallowed.