Iran Press/ Europe: The visitor hit the bar and had two goals disallowed before Franck Kessie scored from the penalty spot to give it a deserved halftime lead.
Roma, who also had a goal chalked off in the first half, quickly responded after the break when midfielder Jordan Veretout curled in his 10th league goal of the season.
But a superb turn and finish from Ante Rebic put the visitor back in front after 58 minutes and it held on despite late pressure from the host.
The victory ends a four-match winless run in all competitions for Stefano Pioli's side, whose hopes of a first Serie A title in 10 years suffered a blow following back-to-back league defeats to Spezia and Inter.
Milan remains second on 52 points, four behind Inter and six ahead of Juventus in third, while Roma remains fifth with 44 points.
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