Reports ahead of the match suggested that Montella faced the sack if his side lost at the Mapei Stadium, but his players rose to the occasion and should have taken the lead in the 39th minute when an unmarked Nikola Kalinic gained possession in the area, and his shot proved too hot for Andrea Consigli to handle.
The Sassuolo goalkeeper tipped the ball behind for a corner, from which Milan did break the deadlock, with Romagnoli beating Consigli to the resulting cross and heading home.
The hosts created few chances of any note during the first period, but almost drew level in the 57th minute when Matteo Politano stung the palms of Milan goalkeeper Gianluigi Donnarumma with a powerful free kick.
Ten minutes later, the visitors killed the game off as Fabio Borini's ball over the top landed at the feet of Suso, and the Spaniard powered into the penalty area and let fly with an effort that Consigli could not prevent from bobbling into the net.
The result sees Montella's men move up to seventh place in the table, four points behind Sampdoria directly above them, while Sassuolo are just two points and one place above the dropzone.