The Belgium international scored his first goal against a top-eight club this season with Uniteds equaliser (39), after Willian fired the dominant champions ahead (32).
Lukaku then crossed for substitute Jesse Lingard to head the winner 15 minutes from time.
The victory sees United reclaim second place in the Premier League from Liverpool and open a potentially crucial six-point gap over fifth-placed Chelsea in the race for the top four.
Chelseas positive midweek performance against Barcelona may have been marred by Lionel Messis away goal but the champions showed no sign of self-pity as they flew out of the traps, the restored Alvaro Morata turning Marcos Alonsos cross onto the bar inside five minutes.
Such was the dominance of the visitors bright, exuberant start that it took United until the 27th minute to register an effort on goal.
First Lukaku had a shot blocked before, a minute later, Alexis Sanchezs toe poke at the end of flowing counter was comfortably saved by Thibaut Courtois.
But just as it seemed the hosts had found a foothold in the game, Chelsea struck as Willian played a one-two with Eden Hazard and bludgeoned a shot through David de Geas defences.
It was no more than Chelsea deserved, as was the lead at the break, but Uniteds only moment of quality denied them that as Sanchez and Martial combined at the end of an intricate move to free Lukaku, who slotted the equaliser home from six yards.
Chelsea continued in the second period from where they had left off, though, as Willian released Morata through on goal on 49 minutes, but Chris Smalling, having initially lunged in, recovered to block the Spaniards shot.
Danny Drinkwaters dipping 20-yard strike drew a fumble from De Gea shortly after, before his counterpart Courtois pulled off a stunning fingertip save to deny Lukaku an acrobatic second on 67 minutes.
But having been denied, Lukaku turned provider, curling a fine cross from the right onto the head of Lingard, who steered his 13th goal of the season beyond Courtois to ensure Jose Mourinho got the better of old foe Antonio Conte.
