The striker drove through the heart of the visiting defence and fired in his 18th league goal of the season to register his club's second fastest Serie A strike since Opta started taking records in 2004.
"We're top of the table and that's a good feeling for us," Lukaku told Sky Italia. "We mustn't give up, last year we lost some of these games but this year we're growing.
"These matches are a chance for us to send a message to the other teams, but the season is long."
Inter coach Antonio Conte added that the club's early Champions League exit has helped their bid to end an 11-year wait to win Serie A.
"It's inevitable that we have to have the ambition to win, that is clear due to the history of the club," he told Sky Italia. "This season I think the disappointment of going out of the Champions League in an undeserved way made us look inside ourselves and understand that we needed to raise the level of everything to try and be more competitive.
"The work is paying off and paid off last year too.
"We are a team that is growing on a collective and individual level, in when to be aggressive off the ball and when to hold the ball.
"We are maturing a lot, but we know well that there are 14 games to go."
Genoa, whose seven-match unbeaten league run came to an end, are 13th with 26 points.