Mauro Icardi's late goal gave the Nerazzurri a glimmer of hope but Domenico Berardi rolled home a late penalty to seal a dramatic but deserved win for the hosts
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.
Mancini handed a first start to Shaqiri since his move from Bayern Munich, with Lukas Podolski also in the first 11, while fellow new arrival Marcelo Brozovic was named among the substitutes.
The visitors started well and twice went close to scoring inside the opening five minutes as Donkor and Nemanja Vidic were both denied from close range – the latter seeing his header well saved by Consigli.
After soaking up the early Inter pressure, Sassuolo opened the scoring with 17 minutes on the clock as a swift counter-attack finished with Zaza firing them into the lead.
The striker collected a pass on the edge of the penalty area and, having held off Vidic, fired a stunning strike into the top corner, giving Samir Handanovic no chance.
Inter struggled to mount a meaningful response to the setback and were punished on the half hour as Sassuolo doubled their advantage.
Having seen his corner cleared, Sansone was given the ball back and, after cutting in from the left, unleashed a ferocious drive off the crossbar and past Handanovic.
Sassuolo felt they should have had two penalties before the break as Donkor twice clashed with Sansone, but both were waved away by referee Paolo Valeri.
With the words of Mancini no doubt ringing in their ears Inter made a better start to the second half, with Shaqiri going closest to halving their deficit with a curling effort from 20 yards.
The visitors were posing much more of an attacking threat as the game passed the hour mark and Shaqiri would have had his first Inter goal had the woodwork not intervened.
As space opened up 20 yards from goal, the Switzerland international fired a low effort past Consigli, but the shot rebounded off the post to safety.
Sassuolo were limited to counter-attacks in the second half, but failed to threaten Handanovic's goal, while Dodo brought a good save out of Consigli.
The Brazilian rose unmarked at the back post to meet Brozovic's cross, but the Sassuolo goalkeeper made a superb reflex save.
However, Inter were given a way back into the match with seven minutes to play as Icardi latched on to a under-hit backpass to pull a goal back.
Sansone's late dismissal for a challenge on Shaqiri slightly soured Sassuolo's day, but they restored their two-goal advantage in stoppage time.
Donkor received a second booking for a foul inside the area on Zaza, and Berardi ensured Sassuolo's third home win of the season from 12 yards.