Leicester prop Martin Castrogiovanni scored the decisive try of a thrilling encounter that saw Les Bleus' Grand Slam hopes destroyed on the opening weekend.
Skipper Sergio Parisse also touched down, but Italy fly-half Luciano Orquera ran the show majestically, kicking a drop-goal, two conversions and a penalty before his replacement Kris Burton's late drop-goal saw the Azzurri beat France for a second successive Six Nations occasion on home soil.
France conjured first-half tries for number eight Louis Picamoles and wing Benjamin Fall, with Frederic Michalak slotting two penalties and a conversion, but they could have few complaints about the result.
Italy survived a frantic finale, holding out despite being under sustained scrum pressure after replacement hooker Davide Giazzon was sin-binned two minutes from time.
But they had heroes everywhere as France threw everything at them, gaining a victory that means they will head to Edinburgh next weekend as favourites to beat Scotland.
France, meanwhile, must beat reigning Six Nations champions Wales in Paris next Saturday or see their hopes of silverware disappear without trace.
Italy made a dream start, breaching the French defence inside five minutes after the tone was set by Michalak sending his kick-off straight into touch.
The Azzurri capitalised on his error by monopolising possession after that, and they produced a magnificent try started by wing Luke McLean's weaving run.
Orquera then sliced open the French midfield defence and Parisse coasted to the line unopposed, rounding off a superb passage of play that gave France immediate food for thought.
Orquera added the conversion and then dropped a goal, but his scoring exploits were rudely interrupted courtesy of a trademark power-surge from Parisse's opposite number Picamoles that opened Les Bleus' account.
A breathless opening quarter ended as it started with another Italian score, this time through an Orquera penalty that gave the home side a 13-5 advantage and left France needing to find far more attacking inspiration than just Picamoles smashing it up the middle.
France almost cut the deficit 13 minutes before half-time after full-back Yoann Huget was driven over Italy's line by his fellow forwards, but television match official Gareth Simmonds ruled that replay pictures were inconclusive.
Michalak booted a penalty awarded after the resulting scrum, and France could sense more points as half-time approached, with midfield gaps beginning to open up.
And the dangerous Huget sparked a move that hauled Les Bleus level, running aggressively from just inside Italy's half and causing sufficient panic in the home team's defence to leave Fall with a clear run.
Michalak kicked the conversion, edging France ahead for the first time and securing a 15-13 interval advantage they just about deserved.
Italy needed to rediscover their impressive early fluency, but technical indiscipline crept into their game, with Michalak landing a long-range penalty after prop Andrea Lo Cicero was penalised by Welsh referee Nigel Owens.
It was typical of the game's unpredictable nature, though, that Italy should score next, and Orquera's role was again pivotal when he delivered a brilliant off-load under pressure to Castrogiovanni.
The powerful forward marked his try by blowing a celebratory kiss to the crowd, and Orquera's conversion put Italy back in front as France captain Pascal Pape limped out of the action.
The 2011 World Cup finalists had it all to do and coach Philippe Saint-Andre began using a star-studded replacements' bench, sending on the likes of scrum-half Morgan Parra and centre Mathieu Bastareaud.
But Burton's drop-goal sealed the deal, meaning a first defeat in five Tests for France and giving Italy an opening weekend Six Nations win for the first time since 2003.
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