Roma were themselves awarded a penalty in the 55th minute and halftime substitute Totti, playing his 25th season at his only professional club, sent Hart the wrong way to score his second goal of the campaign.
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Roma travelled to Turin looking to close a five-point gap to leaders Juventus, but Andrea Belotti’s eighth-minute opener had Luciano Spalletti’s men under pressure from the off.
Inter Milan also capitalised on Roma’s slip despite being held 1-1 at the San Siro by Bologna.
In between, the Torino goalkeeper Hart twice denied his former Manchester City team-mate Dzeko an equaliser for Roma from close range.
Andrea Belotti put Torino ahead before Iago Falque’s penalty made it 2-0.
Falque restored Torino’s two-goal lead 10 minutes later with a shot which took a wicked deflection off the former Tottenham defender Federico Fazio, leaving Roma with 10 points from six games and Torino with eight.
Only legendary striker Silvio Piola, who retired in 1954, has more goals in Italy’s top division with 274.
“I think that’s the best performance I’ve seen from Torino so far, especially given the quality of the opposition”, said Mihajlovic, who joined the club this summer after his sacking by AC Milan.
Roma’s second reverse will prove costlier after Keita Balde and Senad Lulic scored once in each half to secure a 2-0 win over Empoli that moved Lazio up to fifth, behind Roma only on goals difference.
The Rossoblu got off to a good start when Mattia Destro fired them in front on 14 minutes, but Ivan Perisic (37′) responded for Inter before the break to claim a share of the spoils.
But Genoa fell apart in the last 20 minutes as Edenilson and Goran Pandev were sent off before Rey Manaj levelled for promoted Pescara with four minutes left.
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Andrea Ranocchia should have won the match for Inter in stoppage time following a ideal cross from Perisic, but the Italy defender sent a free header wide from six yards out.
