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 were 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 themselves 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 their second win of the season, though, allowing Mourinho to head into the two-week international break in a happier mood.
What does it mean: Mourinho's men back on track
Given the noisy protest flying overhead before kick-off, United showed impressive focus from the off to secure their second win of the season and alleviate some pressure on Mourinho.
The United fans in attendance made their support for the manager clear throughout and would no doubt have been buoyed by a professional and accomplished performance from the players in pink.
Leaders Liverpool might be six points ahead, but Mourinho will have reason to think a corner has been turned in this turbulent start to 2018-19.
Pat on the back: Lukaku lethal but Sanchez superb
Lukaku's first Premier League brace in a year will come as a welcome relief for the striker, especially after he missed an open goal against Tottenham, but Sanchez was the star of the show on Sunday.
Restored to the starting line-up after regaining full fitness, he set up Lukaku's first and played a key role in the move that led to the second, and was a menace to Burnley's backline throughout.
Boot up the backside: Not a day for Mee to remember
Ben Mee will have wanted a good performance given his Manchester City roots, but this was a long way from being one of his best days in a Burnley shirt.
The centre-back misjudged Sanchez's cross to give Lukaku a free header for his first, sat deep to play the United man onside for his second and could have conceded a penalty for bringing him down in the closing stages. A late deliberate handball should have been punished with a red card, too.
United's first game after the international break is another away trip, this time to Watford on September 15, four days before they begin their Champions League campaign against Young Boys. Burnley visit Wolves on September 16.
Key Opta stats:
- Among each of the current Premier League teams, Burnley are on the longest winless run of any side in the competition (nine games – D3 L6).
- Manchester United have won each of their last 24 Premier League matches in which they've scored the opening goal.
- Marcus Rashford was shown his first red card for Manchester United, while also being the first substitute to be sent off for the Red Devils in the Premier League since Marouane Fellaini against Hull City in May 2015.
- At the age of 18 years and 284 days, Burnley's Dwight McNeil became the first teenager to start a Premier League game for the Clarets.
- Manchester United's Marouane Fellaini made his 250th Premier League appearance in this game, while the only other Belgian to make as many in the competition is Vincent Kompany (251).