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Carlo Ancelotti's side moved level with Juventus on 64 points but is still 10 behind leader Inter Milan with just six games left in the season.
Stuttering Juventus drew 1-1 at home with Inter on Saturday and Milan quickly made sure it took advantage - Inzaghi nodding in David Beckham's corner after 13 minutes.
Beckham, returning to the starting lineup after two games on the bench, caused havoc and Inzaghi soon flicked in another of his crosses with fourth-bottom Torino completely powerless.
Inzaghi, 35, converted Ronaldinho's pass in the second half before Kaka netted a penalty, Ivan Franceschini grabbed a consolation and Massimo Ambrosini finished off a good move.
Ancelotti, linked with Chelsea and Real Madrid, had been warned by club bosses that third place was the minimum requirement after limping in fifth last season.
Second spot now looks a distinct possibility with the San Siro meeting with Juventus on May 10 especially important given head-to-head records separate teams on the same points.
Fiorentina wasted the chance to climb into the fourth Champions League qualifying spot when it lost 1-3 at Udinese. Sixth-placed AS Roma moved five behind Genoa after beating Lecce 3-2.