Lukaku could have had a hat trick but allowed teenage forward Sebastiano Esposito to take a second-half penalty. The 17-year-old Esposito converted for his first goal on his first Serie A start.
Esposito was in tears after the match, when he immediately went up to hug his mother.
"It's been a fantastic week, I can't lie, I didn't sleep a lot thinking about how this game would go," Esposito said. "I'd like to thank the team, the coach, the club and Romelu, especially for the goal ... He told me to be sure of myself when I stepped up to strike the ball and to score.
"They're all great people before they're professionals. I found myself in sync with Romelu right away, he's a fantastic person who's always looking to help me, he reassured me a lot by saying that everything would go alright and I'm so happy to have scored."
Esposito became Inter's second youngest goalscorer after Mario Corso. He stepped up to the senior ranks this season and had an impressive Champions League debut in October, despite having only previously played 17 minutes for Inter's first team.
"He was practically a child when he arrived at the training camp this summer, but he's grown massively in the last few months and I had said that at some unexpected point I'd be counting on him in times of need," Inter coach Antonio Conte said.
"He had a big chance today and he certainly made the most of it. He's got a great future ahead of him, both his family and myself will help him, and if he continues like this then he'll only improve."
Inter needed the win to spend the festive period joint top of the table after Juventus won at Sampdoria midweek.
However, Inter was on a three-match winless streak that had seen it slip to second and crash out of the Champions League, and Lukaku's strike partner, Lautaro Martínez, was suspended.
Genoa was second from bottom and had lost its previous six matches at Inter, scoring just once.
Inter broke the deadlock with two goals in quick succession. Lukaku headed in Antonio Candreva's cross in the 31st minute and turned provider two minutes later for Roberto Gagliardini's deflected strike.
Lukaku could have doubled his tally shortly after the hour mark when Inter was awarded a penalty for Kevin Agudelo's foul on Gagliardini. But the Belgium forward gave the ball to Esposito, who celebrated wildly after converting.
Lukaku did get a second in the 71st, taking his tally to 12 in Serie A since his move from Manchester United in the offseason.