Inter were comfortably the better side in Saturday's derby, Marcelo Brozovic's deflected effort early in the second half being followed up by Lukaku's impressive header 12 minutes from time, which led to a celebration between him and the ex-Chelsea boss.
Conte's men also hit the post on three occasions and found Milan keeper Gianluigi Donnarumma in inspired form on a night that could easily have produced more goals.
For Lukaku, the convincing win was a perfect answer to a poor Champions League draw against Slavia Prague on Tuesday.
"My relationship with Conte is very strong," Lukaku said to DAZN.
"He is a coach who really helps me. At the age of 26, I want a coach like this, who motivates and helps me every day.
"I am very happy to be here and to be working with him. This match is very important for us and we are pleased with the three points.
"The game with Slavia Prague was difficult, but we prepared well and gave a strong reply."
Conte, meanwhile, believes there is still much more for Lukaku to give, and he felt Inter's European struggles drove his team on.
"He is 26 years old and arrived here thanks to his potential," Conte said of Lukaku, who reportedly clashed with Brozovic in the dressing room on the night of the Slavia match.
"If we work with him properly, he can become extremely important. He can give even more and I told him that too.
"Brozo and Lukaku are exceptional lads and if anything, I get cross with them because they are too nice. They need to be more clinical on the pitch.
"Quarrels mean there is blood in the veins and a desire to improve. I've been a player, I've argued with team-mates many times and then gone out to dinner with them afterwards. It's not a big deal and they enjoy being together."
Conte added: "We were coming off a performance in the Champions League that left a bitter taste in the mouth and fired us up even more to go into the derby in the right way.
"The lads played a good game in every aspect. It's a deserved victory, I am happy for our fans, because derbies are special games and never easy to win.
"I am happy with our Serie A start, less so in the Champions League, but I do also believe that setback can help bring us back on to the right path.
"A commander can have his plans, but he needs the foot soldiers to go out and execute the orders, so I am fortunate to have very good, disciplined players."