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|Edin Dzeko. (AFP/Cooper NEILL)|
Roma's win meant the top three from last year's campaign all won their opening round of matches.
Champions Juventus, with new superstar signing Cristiano Ronaldo making his debut, won 3-2 at Chievo on Saturday with runners-up Napoli coming back to win 2-1 at Lazio.
But Inter Milan - fourth last season - suffered an early upset after Domenico Berardi slotted in a 27th-minute penalty to give all three points to their rivals from Modena.
"This has complicated everything," said Inter coach Luciano Spalletti.
"Sassuolo did well on the counter-attack, they were compact without giving anything.
"We lost the game because of a penalty and Sassuolo didn't create many chances."
Sassuolo dominated the first half, earning a penalty when Joao Mario brought down Federico Di Francesco in the penalty area.
Inter Milan's top scorer last season Mauro Icardi came close twice, with Croatian Ivan Perisic also sending the ball over the top in the second half.
Earlier Roma's only goal came one minute from time following spectacular play by new Dutch signing Justin Kluivert, son of former Barcelona striker Patrick Kluivert.
The 19-year-old came on as a second-half substitute, running down the right flank, dribbling past past three players to provide the crucial cross which Dzeko lashed into the far top corner.
"It was a great cross, and I scored even better goal!" Dzeko joked.
"Last year I scored one that was maybe a bit better (against Chelsea), but this is among the best three.
"I saw Kluivert on the right and hoped he would cross it to the back post, which is fortunately what he did. He definitely changed the game."
Venezuelan Tomas Rincon had Torino's best chance as he rattled the crossbar on 39 minutes.
Iago Falque thought he had scored for the hosts three minutes after the break but his attempt was ruled offside after a look at the VAR.
Former Paris Saint Germain player Javier Pastore missed a number of chances against his former teammate Salvatore Sirigu in the Torino goal.
Lorenzo De Silvestri limped off after a knock and allowed Ola Aina to also get his first Serie A game just days after arriving on loan from Chelsea, with new Roma goalkeeper Robin Olsen making several saves.
"Edin maybe misses the easiest goals, but has accustomed us to these feats," said Roma coach Eusebio Di Francesco.
"In the second half we suffered, I put Kluivert on to give a little freshness. He has important qualities, today he has shown it."
Parma - returning to the top flight after bankrupty and three consecutive promotions -threw away a two-goal being held 2-2 at home after Udinese scored two goals in four minutes in the second half.
Empoli returned to the top flight with a 2-0 win at home against Cagliari as SPAL won their local derby at 10-man Bologna 1-0.
The opening round of matches involving both Genoa teams were postponed because of the bridge collapse in the port city.
Sampdoria had been due to host Fiorentina and Genoa to travel to AC Milan.
Italian Serie A results:
Torino 0 AS Roma 1
Bolonia 0 SPAL 1
Empoli 2 Cagliari 0
Parma 2 Udinese 2
Sassuolo 1 Inter Milan 0
AC Milan vs Genoa, Sampdoria vs Fiorentina (postponed after Genoa bridge disaster)