Inter, who have two games left, moved above Atalanta into second place with 76 points, four behind Juventus who host Sampdoria on Sunday. Juve, who have three games to play, will wrap up the title if they beat Sampdoria but they have looked shaky recently with one win in five games.
Genoa stayed 17th with 36 points, four clear of the relegation zone although Lecce, who are immediately below them, have a game in hand away to Bologna on Sunday.
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Lukaku broke the deadlock when he got free of Cristian Zapata to head in from Cristiano Biraghi's cross in the 34th minute although Filip Jagiello struck the upright from a free kick shortly afterwards as the Griffins nearly equalised.
Inter coach Antonio Conte has complained repeatedly about the difficulties posed by a hectic fixture list and high temperatures, and his side seemed to be feeling the effects with a sluggish second-half performance.
Inter defender Andrea Ranocchia set up a Genoa counter-attack with a wayward pass out of defence but made amends by blocking Andrea Favilli as he ran on to Nicolo Rovella's through ball.
Yet Inter wrapped the game up in the last 10 minutes. Victor Moses slipped past a defender and his low cross found Sanchez who scored from close range with a shot off the underside of the bar with seven minutes left.
In stoppage time, Lukaku collected a pass out of defence, burst forward, cut inside a defender and drove the ball into the far corner for his 23rd league goal of the season.
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