The big striker, who joined from Everton in the summer, scored in each half, before Anthony Martial and Paul Pogba scored late goals to put the icing on the cake.
The four-goal win puts the Red Devils at the top of the table after the opening weekend of fixtures.
United had the first opportunity in the 20th minute when Juan Mata got in down the left side of the area and pulled the ball back past Joe Hart from the byline towards Lukaku.
However, a combination of three Hammers’ defenders managed to prevent the Belgian striker from scrambling the ball into the net.
United continued to press and were rewarded on 33 minutes when they broke from midfield as Marcus Rashford spotted the run of Lukaku in the area and the potential £90million man smashed in his shot off the inside of the near post.
West Ham’s best chance of the half came right on half time when the ball broke to Edimilson Fernandes on the edge of the area, but his strike was palmed away by David de Gea.
On 50 minutes Rashford burst away down the left channel and into the area, but he curled his right-foot strike wide of the far post.
United went two-up on 53 minutes when Henrikh Mkhitarayan whipped a free kick in from the left and Lukaku got in front of his marker in the six-yard box before guiding a header beyond Hart.
Mkhitarayan had the ball in the net four minutes later but Rashford was offside in the build-up and West Ham were given a reprieve.
Marko Arnautovic went close to scoring on his Premier League debut for the Hammers but his looping header inside the area on 59 minutes bounced off the top of the crossbar.
Lukaku was denied a hat-trick goal on 61 minutes when he was teed up by Mkhitarayan down the right of the area, but Pablo Zabaleta did well to block his goal-bound strike.
Mkhitarayan then tested Hart with a low strike, before Rashford fired an effort off the far post with Hart beaten on 66 minutes.
Daley Blind almost got in on the act on 71 minutes when he met a right-wing cross from Antonio Valencia on the left side of the area and hit a first-time strike which bounced up and off the top of the crossbar.
United grabbed the killer third goal three minutes from time when Mkhitaryan’s through ball fell to substitute Martial inside the area and he coolly slotted his shot past Hart and into the far corner.
There was still time for Pogba to get on the scoresheet on 90 minutes when he collected a Martial pass and curled his effort beyond the dive of Hart from just outside the area.