Lukaku scored twice as Everton beat Leicester 4-2 at Goodison Park on Sunday, revitalising their challenge for a European spot after picking up just one point from their previous two games at Liverpool and United.
The Belgium international, who has been linked with a move away from the club after reportedly turning down a new contract, now has 23 top-flight goals this season, putting him four ahead of closest rival Harry Kane in the race for the Golden Boot.
And he wants his goals to translate to a high league finish for seventh-placed Everton, with Ronald Koeman's men now level on points with Arsenal having played three matches more as United sit three clear of them in fifth.
"At home we have a big advantage with our fans and we are doing really well - now in away games we have to do the same," Lukaku said to Sky Sports.
"We want to fight against Manchester United and Arsenal. We want to put them under pressure. We need to win many more games and in the upcoming tough games we will try to get a result.
"We didn't perform [against Liverpool] and we were really disappointed. We knew from then we had to work much harder to deliver better performances. Now we have won and we have to continue."
Ross Barkley set up Lukaku's first-half header with a perfect cross and the striker is enjoying the link-up play between them.
"We have played together for a long time," said Lukaku. "In this new position he is playing he has got more freedom and can create more chances.
"As a partnership we find each other much better than in the past and hopefully we can deliver more in the upcoming games."
Barkley himself hailed the performance of Lukaku and was thrilled to see Everton get back on track after they were criticised for losing to arch-rivals Liverpool.
"It [criticism] is to be expected," he said. "It is the Merseyside derby and we expect more from ourselves in those big games.
"We went away and worked hard in training. We put a shift in [against United] in midweek and got the three points here which is the main thing – we showed a lot of character.
"It was entertaining. The big man [Lukaku] was on fire and we got the three points which was the main thing. We stuck at it and we knew we had a lot of players like Rom in top form at the minute and quality in the attacking areas."