Mauro Icardi completed his hat-trick from the penalty spot in the final minute, having already twice given Luciano Spalletti's high-flyers the lead.
A brilliant effort from distance by Suso and a Samir Handanovic own goal had Milan within touching distance of a point after a thrilling second-half rally, but Montella was left angered by the decision to penalise Ricardo Rodriguez for impeding Danilo D'Ambrosio in the area.
Despite lavish close-season investment, Milan have now lost four and won four of eight Serie A matches so far this season, and reports this week linked Switzerland's former Lazio boss Vladimir Petkovic with Montella's post.
Nevertheless, Fassone was unequivocal in his backing for the 43-year-old when he spoke to Milan TV after Sunday's match.
"We certainly go along with Vincenzo," he said. "When we are reconstructing our [club] we need to give time and tools.
"Now we see the first positive signs. Montella will absolutely have the time to continue working."
Fassone added: "Derbies are always an amazing thing. This time it turned badly for us and they celebrated with their fans.
"The next time, I am convinced it will be nice for our fans. The second half seemed like a completely different game and, with quality and attitude, we closed Inter in their half of the field.
"[Tenth in the table] is not a position for Milan and we have to roll our sleeves up, but we have received good indications."