The 23-year-old Belgian has now scored 60 league goals for the Merseyside club, level with Duncan Ferguson, and was instrumental in Everton's 2-0 win over Sunderland.
"It's been a good journey. Duncan and I work really hard, first with Roberto Martinez and now with Ronald Koeman. They helped me to improve my game and I'm grateful to them, because without them, I wouldn't have got this far," Lukaku, who has played 129 games for the club, told reporters.
Lukaku, who is also this season's joint highest Premier League scorer with 17 goals, is leading in-form Everton's pursuit of a European place as they sit seventh in the table, unbeaten in nine league games.
"I think it's really important. It has to be the next step for the club. It's down to us to improve and hopefully get into Europe," Lukaku said.
News that makes sense
Your trusted source for staying up-to-date with the world around you. Get free daily news updates and analysis, straight to your inbox.
It was a sentiment echoed by manager Koeman who said his side need to keep chasing Manchester United who are four points above them in sixth.
"There is a gap between us and the teams above us but there are still 12 games to go. You need to continue and get the momentum," the Dutchman told the BBC.
"The team behind us are West Brom and they are in good form, but in two weeks, we play West Brom at home. The team are strong and we have a lot of clean sheets and that makes for a lot of points."
Former Everton boss David Moyes endured a harrowing return to Goodison Park - where he managed for 11 years - and said there was an obvious gulf in class with his bottom-of-the-table side.
"You're got to remember the quality of Everton, where they are and what they're trying to achieve," the 53-year-old said.
"There is a gulf in quality, but the players are giving everything they've got. If we're getting let down anywhere, we don't quite have the quality."
(Reporting by Tom Hayward, editing by Ed Osmond)
