United's £75million record signing struck in each half at Old Trafford to set United on the way to a convincing 4-0 win which was wrapped up in the latter stages by Anthony Martial and Paul Pogba.
After also netting against Real Madrid in the Super Cup last week, Lukaku has begun the season full of confidence and a poor West Ham side certainly had little answer.
"To play in front of such a crowd and for such a great club, it's a dream come true," Lukaku told Sky Sports.
Lukaku finished off a slick move to open the scoring with a powerful finish after 33 minutes and doubled the lead with a header early in the second half.
The Hammers, who were already being outplayed, could not respond and United spurned several more chances before finally adding two more goals in the last three minutes.
Lukaku was denied a hat-trick by a fine block by Pablo Zabaleta and fired another chance over.
Mourinho was also delighted with Lukaku's performance but claimed the 24-year-old was not under pressure to score goals.
He said: "Romelu knows me well. He knows sometimes strikers can play very well for the team without scoring goals. He knows from me there is no pressure at all. But today he was good, very effective."