The Nerazurri had won all three previous meetings between the two sides and scored 15 goals without reply, but that all changed at the Stadio Citta del Tricolore with a comfortable home win piling the pressure on Mancini.
Simone Zaza broke the deadlock with a powerful effort from 18 yards, with Nicola Sansone - who, along with Isaac Donkor, was later sent off - extending the hosts' advantage before half-time.
Xherdan Shaqiri led Inter's charge after the break, but was denied a first Serie A goal for the club when a low shot struck the foot of Andrea Consigli's post.
The introduction of top scorer Mauro Icardi gave Inter late hope with an 83rd-minute strike, but - following a red for Sansone - Donkor was given his marching orders in conceding a penalty that Domenico Berardi converted.
Victory moves Sassuolo above their opponents in the league, Eusebio Di Francesco's side climbing into eighth, while Inter fall to 12th.
