Inzaghi saw his side take the lead through Mattia Destro early in the second half, as Milan looked set to bounce back from disappointing draws to Chievo and Verona.
However, substitute Joaquin supplied the assist for Gonzalo Rodriguez's equaliser before scoring the crucial second to bolster his side's European aspirations.
Milan, who would have leapfrogged rivals Inter with a win, are left with plenty of work to do to ensure European football.
"I have a contract and will continue to work away quietly," Inzaghi told Sky Sport Italia.
"There have been many difficulties, but after tonight we can get back to playing good football. I go along from day to day and I hope to stay for a long time on the Milan bench.
"If they want to keep me, great, otherwise I will go ahead on my way.
"I want to this job for hopefully 30 years. It is clear that Milan is a club that has given me so much and I hope to stay as long as possible.
"I work for my future and for the future of Milan. If they will keep me great, if not I will go on my way, always head-on, with professionalism, with great desire."
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