The forward struck twice, while Giampaolo Pazzini also got on the scoresheet to boost the coach's chances of keeping his job next season
El Shaarawy was making his first start for four months and made up for lost time by scoring a goal in each half as Torino's hopes of securing a Europa League spot were ended at San Siro.
Pazzini also scored from the penalty spot after Cristian Molinaro had been sent off for bringing down Marco van Ginkel to take his goal tally in the top tier to three figures in what is expected to be his final home game for the club.
Milan were already down to 10 men themselves, as Cristian Zaccardo was dismissed for a professional foul on Molinaro late in the first half - which was their fifth red card in as many games.
Inzaghi's men had lost four of their last five games before Sunday's clash and despite an encouraging display in their penultimate games of the season, the coach is expected to be dismissed with Sevilla's Unai Emery and Carlo Ancelotti linked with the role.
The recalled El Shaarawy took only 18 minutes to produce the type of quality that Milan have been lacking this season by opening the scoring with a clinical finish.
The Italy international found space just inside the penalty area and controlled a cross from Zaccardo in an instant with his first touch before firing into the bottom-left corner of the net with his second.
El Shaarawy could have doubled his tally eight minutes before half-time when he seized upon a loose pass and played a one-two with Pazzini before snatching at his shot, which was well off target.
Milan's indiscipline has been a major problem in recent weeks and there was a sense of deja-vu they were reduced to 10 men once again three minutes before half-time.
Zaccardo was the guilty party on this occasion, upending Molinaro after Amauri flicked the ball into the path of the marauding defender to prompt referee Paolo Valeri to reach for the red card.
Molinaro was on his way down the tunnel himself 11 minutes after the break, though, after he clattered into Van Ginkel to give Pazzini the opportunity to reach his landmark.
The former Inter forward made no mistake, driving his spot-kick down the middle and racing away to celebrate by holding up a shirt with 100 on to mark the occasion on what would appear to be his San Siro farewell.
El Shaarawy then ensured the spotlight was back on him by scoring his second goal of the night after 65 minutes, exchanging passes with Van Ginkel before calmly placing the ball beyond Padelli.
The striker was given a great reception when he was replaced a couple of minutes later after an impressive display and Inzaghi must have been wishing he had been able to call upon the 22-year-old on a regular basis in what has been a miserable season.
Christian Abbiati pulled off a fine late save to deny Omar El Kaddouri in what is also likely to be his last home game for Milan, who were able to give their suffering supporters something to shout about at the expense of Torino.