Lukaku scored his first goal for Manchester United in a 2-1 win over Real Salt Lake on the second game of their US tour.
The striker, who joined United from Everton in a £75m deal, said he had already added to his game since being reunited with his former Chelsea boss Mourinho.
Lukaku said: "He (Mourinho) knows what I can bring to the team and he's someone who is pushing me."
"I push myself too, but he makes me a better player."
"I think I learned a lot from him in the last week and I'm glad I can work with him."
"He said I had to keep doing what I'm doing."
"I'm happy for my goal. There's still a lot of work to do, as a team we know we need to be much better.
"But I'm glad I'm here because you really see it's a big club and that was the challenge I wanted."
"I'm very happy now. This is the move I wanted."
"I have to make sure that I stay focused and have the same ambition I've always had and work hard, that's the key to success."