Jose Mourinho's men secured a morale-boosting win to reduce some of the pressure on their manager despite Paul Pogba's missed penalty and Marcus Rashford's red card in the space of two second-half minutes.
A banner protesting against executive vice-chairman Ed Woodward was flown over the ground before kick-off, but United was not distracted from the job at hand in the early exchanges, with Jesse Lingard missing three chances in a dominant opening spell.
Burnley grew into the game but found itself behind 27 minutes in, when Lukaku jumped just high enough to head a fine Alexis Sanchez cross past Joe Hart from six yards out.
It needed a fine save from Hart to deny Lukaku a quick second after a swift break involving Sanchez, but the Belgium striker smashed home from point-blank range to make it 2-0 before half-time after Lingard's shot hit first Charlie Taylor and then Ashley Westwood.
Paul Pogba wasted the chance to make it 3-0 when Hart comfortably saved his rather tame penalty, and Burnley hopes of a comeback were increased two minutes later when substitute Rashford was sent off for pushing his head into Phil Bardsley's after he appeared to kick out at the forward.
Sean Dyche's men pushed forward with the man advantage but Lukaku should have killed off the game, the forward wasting two opportunities when put clean through one-on-one with Hart.
United held firm for its second win of the season, though, allowing Mourinho to head into the two-week international break in a happier mood.