The 24-year-old - signed from Everton earlier this summer - scored either side of half time before Anthony Martial and Paul Pogba contributed late efforts to consign West Ham to a heavy defeat on the opening weekend of the season.
After a bright opening from both sides, the first opportunity fell to Henrikh Mkhitaryan inside the penalty area but after capitalising on Arthur Masuaku's poor header, the Armenian blazed over from 18 yards.
As the first half approached the midway point, Juan Mata was denied by Joe Hart from close range, before the Spaniard failed to make the most of another opportunity from a similar position where he unselfishly tried to pass to Lukaku instead of getting another effort on goal.
However, shortly after the half-hour mark, United had their opener when a brilliant through-ball from Marcus Rashford found the run of Lukaku, and the summer signing lashed the ball into the net from 12 yards with his shot going in via the inside of the post.
The home team deserved their lead but West Ham had remained competitive throughout the first 45 minutes, and they could have equalised in added-on time when Edimilson Fernandes warmed the hands of David de Gea after some sharp work from Andre Ayew.
Soon after the restart, Rashford should have doubled United's lead after running onto a superb through-ball from Paul Pogba but despite outpacing the defender, his shot was directed well wide of the far post.
However, it was not long before Lukaku added his second of the match, with the forward rising above the West Ham defence to head Mkhitaryan's free kick into the bottom corner.
United were dominating at this stage but shortly before the hour, West Ham came within inches of halving the deficit when Marko Arnautovic was found in the penalty area, but the winger's header clipped the bar and went over.
That opportunity ensured that there was no complacency from United and three times in the space of 10 minutes they came close to another goal, with Mkhitaryan seeing a low shot saved by Hart and Rashford striking the inside of the post from 20 yards.
During the closing stages of the contest, chances were few and far between but with three minutes left, United capped off a relatively comfortable afternoon with a third goal from Martial, who found the bottom corner after collecting a pass from Mkhitaryan.
Given their endeavour, West Ham did not deserve to be three goals behind but it got worse in the final minute of normal time as Pogba curled a shot past Hart from 20 yards out.
It capped a fine performance from Jose Mourinho's side and if this display is anything to go by, their rivals will face more competition in the race for the Premier League crown this season.