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.
Inter's title challenge has faded over the last few weeks after they let several games slip from their grasp, leaving manager Antonio Conte appreciative of his team's efforts while bemoaning some of their results.
"Seventy-six points and the best defence? In the last few weeks, and I don't know why, the glass has been seen as half empty," said Conte. "But these numbers speak clearly and are comforting. These lads are doing very good things. Of course, we can and must improve.
"They're not giving up although some games, like the one against Fiorentina, were disappointing," he said referring to Wednesday's 0-0 draw. "I think there are years in which you sow little and reap a lot and others in which the opposite happens."
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.
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.
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.