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Milan were next to threaten, as Giacomo Bonaventura fired a strike from range inches wide of the far post.
The keeper was shown a straight red card and gave away a penalty in the process, which was duly converted by Bacca.
Despite their numerical disadvantage, Sassuolo got back on level terms early in the second half when Domenico Berardi curled a free kick beyond the reach of Donnarumma and into the far corner.
Milan quickly looked to restore their lead and Alessio Cerci saw an effort saved, before Riccardo Montolivo dragged wide from the edge of the box.
The Rossoneri kept pressing for a winner and with four minutes remaining substitute Adriano found space inside the box to meet Bonaventura's corner, before powering a header into the net to seal the points.