With the result, United regain second place from Liverpool and open up a six-point gap back to Chelsea in fifth.
Alvaro Morata was the match-winner in the reverse fixture at Stamford Bridge in November and came within inches of opening the scoring on four minutes.
After a quick start to the game, both teams retreated and were content to sit back and try to hit each other on the counter-attack, which limited the amount of goalmouth action before the half-hour mark.
That approach nearly worked for United when Anthony Martial broke away down the left and picked out the run of Sanchez, who could have shot first time but subsequently poked tamely at Thibaut Courtois.
Chelsea showed more conviction when they had their opportunity, with Willian starting the break on the edge of his own box and finishing it with a thumping shot through De Gea at his near post on 32 minutes.
Lukaku quickly levelled the scores after United had constructed the best move of the match, passing the ball between Sanchez and Martial in a tight area before the Belgian swept home from eight yards.
At the other end, Courtois had to be at his best on 67 minutes to get a fingertip to an acrobatic Lukaku volley, which was set up by Sanchez with one of his few contributions on an otherwise quiet afternoon.
Thankfully for the Red Devils, Lukaku stepped up to answer some questions over his performances against top-six opposition this season, adding to his first goal in seven games against United's closest rivals by crossing for Lingard, who had escaped the attention of Andreas Christensen, to head past Courtois.
Conte had already substituted Eden Hazard in his attempts to seal a draw, but the Blues did have a couple of chances to rescue a point as De Gea saved from Willian's volley and Morata had a goal disallowed.
The Spaniard squeezed a left-footed shot into the bottom corner from a lofted Cesc Fabregas pass, but was fractionally offside, small margins which could be crucial in the quest for Champions League qualification.
MAN UNITED (4-3-3): De Gea; Valencia, Lindelof, Smalling, Young; Pogba, Matic, McTominay; Martial (Lingard, 64'), Alexis (Bailly, 81'), Lukaku
CHELSEA (3-4-3): Courtois; Rudiger, Azpilicueta, Christensen; Moses (Giroud, 78'), Drinkwater (Fabregas, 81'), Kante, Alonso; Willian, Hazard (Pedro, 73'), Morata