The Toffees closed the gap on the Champions League places to just six points courtesy of goals from Idrissa Gueye and Lukaku, with the Belgian drawing level alongside Duncan Ferguson as the club's leading scorer in Premier League history.
Ronald Koeman's side have now picked up a league-high 17 points in 2017, and Lukaku is hopeful of continuing that form and booking their place in Europe next term.
"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," he told reporters.
"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. I was always sure I would play [after a recent calf injury]. My rehab coach in Belgium worked really hard to get my body stronger. When I got back, I had no problem.
"Jermain Defoe was unlucky [when he hit the crossbar]. He is a top-quality striker. He has shown year in year out what he can do. His bad luck was our good luck. We have to win as many games as we can. Next week, we have a difficult game at Spurs. If you look too far ahead, you might not be focused on the game that's coming."
Everton are now unbeaten in their last nine Premier League games since December 19.