All the talk pre-match was about Jose Mourinho's future at the club, with United having picked up just three points from a possible nine so far this season but United flew out of the blocks early on.
Jesse Lingard looked sharp early on, curling an effort just wide form the edge of the box before the again impressive Luke Shaw called Joe Hart into the first of many saves with a low drive from range.
United dominated for long spells and got their reward with 27 minutes gone as Lukaku bagged his first of the game.
Phil Bardsley, who would later become United's public enemy number one, gave Alexis Sanchez too much space out on the left and the Chilean lofted a cross to the far post, allowing Lukaku to peel off the back of Ben Mee and power home a header.
Lukaku went close again nine minutes later after being slipped in by Sanchez, but Joe Hart won the duel this time as he stayed big to keep out the striker's attempted chip.
The visitors needn't have worried though, as their dominance told again on the stroke of half-time as Lukaku doubled his and United's tally.
Lingard was afforded the space to crack a shot off from the edge of the box, with his effort striking both Charlie Taylor and James Tarkowski before dropping perfectly for Lukaku to sweep home right-footed.
The brace was only Lukaku's second in the Premier Legaue for Manchester United, with his last coming against West Ham on his debut for the club.
September 2, 2018
United continued to be the better side after the break, with Shaw seeing another fierce strike beaten away by Hart at the near post.
With 68 minutes gone, Burnley looked to have handed United the perfect way to seal the victory as Aaron Lennon felled substitute Marcus Rashford in the box, only for Hart to make another fine stop from Paul Pogba's spot kick.
The Frenchman struck his effort well, but the former Manchester City man dived away to his right to keep the effort out.
United could have been forgiven for thinking that everything was going wrong for them just three minutes laster as Rashford was dismissed for thrusting his head into the face of Bardsley after a clumsy challenge from the experienced defender.
September 2, 2018
Burnley piled on strikers in their search for a route back into the game, but despite their one-man deficit the away side continued to threaten, with Lukaku again being denied by a fine Hart save after being slipped in behind.
Lukaku wanted another penalty when he went down under the challenge of Ben Mee after rounding Hart late on, but Jon Moss waved away his appeals.
For all their attacking players on the pitch, Burnley failed to create a real gilt-edged chance, though David De Gea did make one diving save to keep out a Sam Vokes header in the closing stages.
The win sees United move up to 10th, whilst Burnley go into the international break sitting 19th with just a point to their name.
September 2, 2018