Marcelo Brozovic, with an exquisite curling shot, and Stefano Sensi gave Inter a 2-0 halftime lead before Lukaku, signed from Manchester United this month, opened his account on the hour.
Antonio Candreva completed the rout with a stunning strike from 35 metres.
Chinese-owned Inter appointed Conte, who previously won three Serie A titles with Juventus and one English Premier League title with Chelsea, to replace Luciano Spalletti in the close season in a bid to end eight seasons without a major trophy.
As expected, there was no place for forward Mauro Icardi who has been told he is not part of Inter's plans but has yet to negotiate a move away from the club.
"I am very happy and satisfied; I like working with these guys who have always shown great willingness," said Conte.
"Romelu has entered the Inter world with great humility, he is ready to work for the team."
Lecce, back in the top flight after a seven-year absence, made a bright start, forcing Brozovic to make an acrobatic clearance and then Milan Skriniar to clear away Gianluca Lapadula's effort with goalkeeper Samir Handanovic beaten.
But Inter quickly began playing the whirlwind football associated with Conte and Lautaro Martinez hit the crossbar with a free header from a Candreva cross.
Brozovic broke the deadlock in the 21st minute when he curled a delightful shot into the far corner from the edge of the area.
Sensi, making his debut after his move from Sassuolo, added the second three minutes later, scoring with a low shot which went through a defender's legs.
As Inter continued to pour forward after halftime, Lukaku snapped up a rebound after Martinez's shot was parried by goalkeeper Gabriel to add the third.
Lecce finished with 10 men after second-half substitute Diego Farias, making his debut after a move from Cagliari, was sent off for a reckless tackle from behind on Danilo D'Ambrosio.
Candreva's dipping, long-range effort capped a near-perfect start for Conte's side and sent Inter fans home with hope that this season they can finally challenge Juventus for the title.
(Writing by Brian Homewood; editing by Clare Fallon)