Meanwhile, Frank de Boer, one of eight coaches making their debuts at a new club on Sunday, got off to a losing start when Inter Milan lost 2-0 at Chievo while last season's runners-up Napoli hit back from two goals behind to draw 2-2 at Pescara.
Donnarumma saved Andrea Belotti's penalty in the 96th minute to give Vincenzo Montella a winning start as he became the sixth man to sit in the Milan hot seat in the last two-and-a-half years.
The match also brought Milan face to face with Sinisa Mihajlovic, the coach they sacked in April and who gave Donnarumma his professional debut as a 16-year-old last season.
“If I’d have known he was going to save the penalty, I wouldn’t have given him his debut," said Mihajlovic, who then made a bizarre comment about Belotti's penalty.
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“I smiled because I knew he was going to miss and I’m almost always right," said Mihajlovic, who was taking charge of Torino for the first time.
"I went to reassure him afterwards, that these things happen and told him that the next time we get a penalty he won’t be taking it.”
Colombia forward Bacca scored three times between the 38th and 62nd minutes, the third a penalty, while Belotti briefly pulled Torino level just after halftime.
Milan led 3-1 going into stoppage time, yet nearly contrived to throw it away.
Daniele Baselli pulled one back, then Belotti was fouled inside the area by defender Gabriel Paletta who was given a second yellow card and sent off. But Belotti was foiled by Donnarumma.
Two second half goals from Valter Birsa gave Chievo the points against Inter, who hired Dutchman De Boer two weeks ago after Roberto Mancini left by mutual consent.
Birsa wrong-footed two defenders to score in the 49th minute and the Slovenian raced past Gary Medel to wrap up the points for the Flying Donkeys with an emphatic finish with nine minutes left.
Promoted Pescara stunned Napoli when Ahmed Benali and Gianluca Caprari gave them a 2-0 halftime lead, but Dries Mertens came off the bench in the second half and almost immediately scored two goals in four minutes to equalise.
Napoli, shaken by the departure to Juventus of last season's Serie A top scorer Gonzalo Higuain, then thought they had been awarded a penalty but the referee changed his mind after consulting his assistants.
Marco Giampaolo's first game in charge of Sampdoria took him straight back to his previous club Empoli, now coached by Giovanni Martusciello, and a first-half goal by Luis Muriel gave him a winning return.
Gian Piero Gasperini's first game with Atalanta ended in a 4-3 defeat at home to Lazio, who re-hired Simone Inzaghi during the summer after eccentric Argentine coach Marcelo Bielsa quit after two days.
Genoa, in their first game under Ivan Juric, beat promoted Cagliari 3-1 while Juric's former club Crotone, playing their first-ever Serie A game, lost to a late Mattia Destro goal at Bologna.
(Writing by Brian Homewood; Editing by Rex Gowar)
