The Belgian striker, signed by Inter for a club record 80m euros (£74m) from Manchester United on 8 August, tapped in the third goal in the 60th minute.
Antonio Candreva’s 30-yard drive completed the victory after Leece’s Diego farias had been sent off for a nasty looking challenge.
Lukaku, who hit 42 goals in 96 games for United after joining from Everton for £75m in July 2017, has now scored on his league debut for each of his past four clubs.
Inter boss Antonio Conte, who replaced Luciano Spalletti in May, had identified Lukaku as a key summer target and was pleased with his star signing.
“Romelu has entered the Inter world with great humility,” Conte said. “He is ready to work for the team.”