Lukaku scored twice in Tuesday's 4-1 victory over Cagliari to take his season tally to 18 goals across all competitions for Inter since his club-record arrival at the start of the campaign.
The Belgium international opened the scoring after just 21 seconds, while he also headed in Inter's third in the second half at San Siro, where the Nerazzurri advanced to the quarter-finals.
"I must thank my team-mates, the coach, the backroom staff and the fans. Everyone has helped me from day one," Lukaku, who has scored six goals in his last four games, told Rai.
"Inter were my best option. I can grow here and I can give my contribution. My role is to help the team in any way and I know that the fans like this. I must keep this up.
"Obviously, scoring is the objective, however, I'm not just thinking about myself, but also the team. I'm here to achieve this and I hope to win something. However, for now, I'm only focused on Lecce."
Inter head coach Antonio Conte, who wanted to sign Lukaku during his time at Chelsea before finally getting his man this season, said: "You know that I don't like to talk too much about individuals. Romelu is proving his worth and he can still keep improving."
On the team following Saturday's 1-1 draw with Atalanta, which saw Inter drop two points behind Serie A champions Juventus in the race for the Scudetto, Conte added: "I'm really pleased, it was a great response from the lads. Even from those who've played less of late and those who were coming back from injury.
"I'm happy for [Nicolo] Barella, who is recovering after his injury and he was one of our starters today, as was [Alexis] Sanchez. He is an important player for us and slowly he'll get back to his best. I'm also pleased for Borja Valero, who proved himself once again today, as well as [Valentino] Lazaro, [Federico] Dimarco and [Andrea] Ranocchia, who haven't had many chances so far but all played really well.
"There's real harmony among this group, they are a great bunch of lads that want to do well. I'm really pleased to be working with them. We rounded off an important first half of the season in the league, picking up 46 points despite our difficulties and they have now left us in a strong position.
"The second half of the campaign won't be easy but we've started on our journey and we want to keep growing, maintaining the right amount of patience to bring everything together."