The Belgium international gave the Nerazzurri the lead 34 minutes into the clash when he headed past Mattia Perin after a cross from Cristiano Biraghi.
After Alexis Sanchez doubled the visitors' lead late in the second half, Lukaku charged into the box to meet a pass from Marcelo Brozovic and fire into the net.
Lukaku has now scored 15 away goals in the Italian top flight and is the first Inter player to reach that tally in 70 years, equalling the record set by Stefano Nyers.
15 - Romelu #Lukaku has scored 15 goals away from home in the current Serie A season, equalling the record of goals for an #Inter' player in a single top-flight campaign (Stefano Nyers in 1949/50). Fantastic.#GenoaInter pic.twitter.com/e0c0ijqCcA— OptaPaolo (@OptaPaolo) July 25, 2020
With six goals from the nine league games played since mid-June, he has taken his tally to 23 in Serie A and 29 in 44 appearances in all competitions.
Saturday's victory extends Inter's unbeaten run in the Italian top flight to six games and moves them up to second in the table with two matches left to play.
Lukaku is happy with his performances since his summer move, but admits his side have slipped up on too many occasions this term.
“On a personal level, I am happy, but I also say that as a team we can do so much more,” he said to Sky Sport Italia.
“We have a good squad, it’s not easy for us to be in second place, because we want to win and go as high as possible. I want to help Inter achieve that.
“We dropped too many points along the way, but we are young and eager to improve. Now we want to do well in these final rounds and then the Europa League.”