The Belgian registered his first Premier League brace since his United debut at the start of last season, opening the scoring shortly before the half-hour mark and adding a second right at the end of the half as the Red Devils bounced back from successive defeats.
Burnley avoided further damage when Paul Pogba saw his penalty saved by Joe Hart, and they were handed a potential lifeline moments later as Marcus Rashford was shown a straight red card following a clash with Phil Bardsley.
However, the lacklustre Clarets could not build on their numerical advantage and remain winless this season, having now picked up just one point from their opening four games of the campaign.
A plane carrying a banner reading 'Ed Woodward: A specialist in failure' flew over the stadium before the match, and United's failure to bring in a new defender during the summer was highlighted once again as Jose Mourinho named his fourth different centre-back partnership.
Mourinho also welcomed the fit-again Alexis Sanchez back into his side, and the Chilean teed up Jesse Lingard for an early chance as the England international curled his effort inches past the post after only three minutes.
Luke Shaw then drew the first save from Hart moments later as United made a positive start to the match, clearly keen to make amends for successive defeats at the hands of Brighton & Hove Albion and Tottenham Hotspur.
Lingard was the main protagonist in the opening exchanges, though, and he saw one looping effort clawed away by Hart before getting a 25-yard effort all wrong in the 10th minute.
Burnley did eventually begin to settle into the game and stem the tide of United chances, but the opening goal arrived when the hosts were enjoying their best spell of the game in possession.
It was their uncharacteristic lack of intensity which was the main issue for Sean Dyche's side, and that was typified by the first goal when Sanchez was allowed the space to pick out a pinpoint cross to the back post, where Lukaku powered his header past Hart having peeled away from his marker.
The Belgian should have had a second less than 10 minutes later when he played a one-two with Sanchez on his way into the box, but Hart was quickly out to thwart the striker as he tried to lift his finish over the onrushing keeper.
Lukaku would not have to wait too much longer to double his personal tally, though, putting his side in complete control right on the stroke of half time when he reacted quickest to a loose ball and swept his finish past Hart.
Dyche sent his players out early for the second half following a disappointing display in the opening 45 minutes, but little changed after the interval and some more lax defending almost gifted Sanchez a goal until James Tarkowski recovered to make an important challenge.
Sanchez had another sight of goal shortly before the hour mark when he slipped while making contact with the ball, and the sum of Burnley's response was a wayward header from Chris Wood while under pressure from Chris Smalling.
Smalling squandered a chance of his own at the other end shortly afterwards, while Shaw also tested Hart from a tight angle, but United's best chance of a third goal came from the penalty spot after Aaron Lennon was adjudged to have fouled Rashford.
Pogba took on the spot-kick duties despite Lukaku being on a hat-trick, and the World Cup winner saw his penalty saved at the end of his trademark long run-up as Hart guessed right to deny the France international.
The penalty save sparked a flurry of activity in the final 20 minutes, and just two minutes after winning the spot kick Rashford was given his marching orders courtesy of a straight red card from referee Jonathan Moss.
The striker, who had only been on the field for 10 minutes, reacted angrily to a challenge from Bardsley and butted heads with the former United defender, giving himself little grounds for complaint as he trudged off the field.
Burnley could not build on the natural momentum that comes with a penalty save and red card in such quick succession, though, and it was United who continued to look the most likely to add to the scoring, with Lukaku being denied his hat-trick by another important save from Hart moments later.
Sam Vokes could not make the most of a rare Burnley chance at the other end, and Lukaku should have put the game to bed once and for all just four minutes later when he raced clean through on goal before taking the ball past Hart. However, Ben Mee recovered and made a crucial challenge on the hat-trick-hunting forward, who felt as though he should have had a penalty.
The few goalscoring opportunities Burnley did create came via the aerial route, with Vokes twice seeing headers denied by David de Gea in the closing stages, while Matej Vydra also sent a header flashing past the near post in stoppage time.
There were also late sights of goal for Pogba and Ander Herrera as United comfortably saw out a deserved victory, recovering from their worst start to a season since 1992 and keeping their first clean sheet of the campaign in the process.
Burnley, meanwhile, remain in the relegation zone following a third successive Premier League defeat as their winless start to the season drags on.
BURNLEY (4-4-1-1): Hart; Bardsley, Tarkowski, Mee, Taylor; Lennon, Cork, Westwood, McNeil (Barnes 80'); Hendrick (Vokes 58'); Wood (Vydra 84')
MAN UTD (4-3-3): De Gea; Valencia, Lindelof, Smalling, Shaw; Fellaini, Matic, Pogba (Bailly 92'); Lingard (Herrera 76'), Lukaku, Sanchez (Rashford 61')