A Mario Balotelli corner was headed home by Alex in the 18th minute but Mario Mandzukic scored an equaliser for Juventus at San Siro after being played in by Alvaro Morata.
However, Juventus managed to find a winner through Paul Pogba in the 65th minute, extending their lead atop the Serie A standings and stretching their unbeaten run in the league to 22 games.
Mihajlovic said he could not fault his players for their performance despite the defeat seeing them lose ground on the European places.
"The result is what makes me angry, because we did not deserve to lose this game. I could not have asked the players for more," Mihajlovic said.
"Milan have their defects, certainly, but we really haven't been lucky this season.
"This evening it wasn't just three points at stake, but also our pride and our honour.
"We didn't want to lose face again. We can hold our heads high, having played on level terms with Juve.
"I saw hunger, character and intensity. [Juventus goalkeeper Gianluigi] Buffon needed two or three great saves and he made the difference.
"We must continue with this determination, as the psychological aspect has always been our real problem."
