Lingard was in the right place after 75 minutes to decide a match in which Chelsea had made by far the more purposeful start.
The visitors were within a whisker of opening the scoring after five minutes when Eden Hazard found Marcos Alonso, whose first-time cross was volleyed against the bar by Alvaro Morata.
Alexis Sanchez saw a free kick held by Thibaut Courtois as United stirred after 11 minutes, but there was little in the way of goalmouth action in a cagey start.
Morata was foiled by a Chris Smalling challenge as he tried to burst into the area before, at the other end, Anthony Martial saw a threatening low cross scrambled away.
Another Martial cross provided a chance for Sanchez to strike at goal only for the former Arsenal man to get his shot all wrong, and United were made to regret that as Chelsea took the lead after 32 minutes.
Willian started the move, breaking from just outside his own area before feeding Hazard and then running on to the Belgian's return pass to blast past David De Gea.
But Jose Mourinho's side were level five minutes later as Sanchez and Martial set up Romelu Lukaku to level against his former club with a low shot.
Alonso shot over with the game opening up as the first half entered its closing stages, and the second began with Morata getting away from Smalling but seeing an angled strike blocked.
Ashley Young then tried to slide a pass through to Lukaku inside the area as the home fans urged their forwards to put more pressure on the visiting defence.
With 58 minutes gone, Danny Drinkwater tried his luck from distance, with De Gea making a meal of it but seeing the rebound hacked to safety.
Willian fired over moments later before Mourinho brought on Jesse Lingard for Martial as he looked to add impetus.
After 68 minutes, Lukaku almost conjured another fine goal when his acrobatic effort was superbly tipped over by Courtois.
But Mourinho and United did not have to wait long for Lingard to make the decisive impact.
With 75 minutes gone -- moments after De Gea had almost spilled a low strike from Willian -- the United substitute produced a neat header from Lukaku's excellent cross to beat Courtois and seal the points for United.