Torino claimed that Juventus midfielder Sami Khedira fouled Afriyie Acquah as he won possession to set up the move which ended with Mario Mandzukic scoring in the 67th minute.
Torino's players persuaded the referee Daniele Doveri to consult VAR but were livid when, having studied the replay on a pitchside monitor, the official refused to change his mind.
Mihajlovic was sent off after remonstrating furiously and other players also surrounded Doveri.
Mihajlovic refused to speak to the media after the game and instead sent his assistant Attilio Lombardo to answer questions.
"Acquah was knocked over," Lombardo said. "What's the VAR for? If he's looking at it on the television, I don't see how the referee could see it differently to the way we did."
Juventus coach Massimiliano Allegri said: "I haven't had another look and I don't want to comment. We could have scored the second goal before that."
Juventus, who rested forward Gonzalo Higuain and fielded Mandzukic as a lone striker, took the lead after 15 minutes when Douglas Costa volleyed home after the ball had ricocheted around the penalty area.
Torino had their best chance of the match shortly afterwards as a long clearance by Vanja Milinkovic-Savic found M'Baye Niang and the former AC Milan and Watford forward blasted his shot against the post.
Apart from that, Torino, beaten 4-0 by Juventus in the league earlier in the season, did not look like improving on a run which has seen them win only one of the last 25 meetings between the Turin neighbours.
Juventus playmaker Paulo Dybala, scorer of two goals in the 3-1 Serie A win at Verona on Saturday, was again impressive, suggesting that he has left a poor spell of form behind him.
Juventus will meet Atalanta in their two-leg semi-final while AC Milan meet Lazio in the other tie.