Goals from the Belgian and Alexis Sanchez inside 20 minutes gave the Red Devils the ideal start as they made light work of Carlos Carvalhal's much-improved outfit.
United were immediately on the front foot and Lukaku brought up his milestone within five minutes as his deflected shot from 15 yards out flew past Lukasz Fabianski.
Having played a part in the first, Sanchez added his own name to the scoresheet, for just the second time in United colours, with a sweeping finish into the bottom corner after Jesse Lingard's pass had rolled under the foot of Federico Fernandez.
Lingard himself had a pair of chances either side of Sanchez's strike but could not build on his goal for England during the international break as he twice shot wide when well placed on the edge of the box.
Sanchez was the brightest he has been since his high-profile arrival from Arsenal and could have increased United's lead either side of the break, firstly swinging and missing from an Ashley Young knockdown and then failing to keep his header from Lukaku's layoff underneath the crossbar.
Swansea did not manage a shot on target throughout the first period, but the half-time introduction of Tammy Abraham made them more of an attacking threat, and David de Gea had to be at his best to prevent the substitute from pulling the visitors back into the contest.
After turning skilfully away from Victor Lindelof, Abraham unleashed a powerful shot which De Gea kept out with a strong right hand, and the Spaniard reacted quickly on his line to deny the on-loan Chelsea striker moments later, when he met Martin Olsson's cross with his left foot.
Although their quick start appeared to set up an opportunity for a more emphatic victory, United did not make a concerted push for more goals late on, with Lukaku spurning their best chance for a third goal on 79 minutes when his shot was blocked by Fabianski from point-blank range.
Nevertheless this routine home win was enough to lift Jose Mourinho's side back above Liverpool into second in the table, after the Reds had briefly leapfrogged their rivals with victory over Crystal Palace at lunchtime.
MAN UNITED (4-2-3-1): De Gea; Valencia, Smalling, Lindelof, Young; Matic, Pogba; Mata (McTominay, 90'), Lingard (Herrera, 76'), Sanchez (Rashford, 75'); Lukaku
SWANSEA (5-4-1): Fabianski; Naughton, Van der Hoorn, Fernandez, Mawson, Olsson; Dyer (Abraham, 46'), Ki (Carroll, 46'), King, Clucas (Routledge, 59'); Andre Ayew