Katusha rider Kristoff timed his effort perfectly to win with a comfortable margin to leave Slovakian Sagan, who looks set to win the green jersey for points classification, still without a win on the Tour this year.
French champion Arnaud Demare took third place and Italian Vincenzo Nibali retained the overall leader's yellow jersey.
"I've always dreamt of winning a stage on the Tour de France," Kristoff, who snatched his major victory when he won Milan-San Remo this year, told a news conference.
"It's the second biggest win of my career."
A five-man breakaway was reduced to four when Spain's David de la Cruz went down hard on a bend and pulled out with a suspected broken collarbone.
Then they were just two, Dutchman Sebastian Langeveld and Simon Clarke of Australia on their own ahead of a bunch led by Europcar and Giant-Shimano who were looking to set up their sprinters for the finale.
Europcar played another card, however, with Frenchmen Perrig Quemeneur and Cyril Gauthier attacking to join Clarke after the Australian had shaken off Langeveld.
But the peloton led by Giant-Shimano reined them in with a few kilometres left.
GREIPEL CRASHES
German Andre Greipel, one of the pure sprinters who had managed to go through the two climbs near the finish, crashed after tangling with Frenchman Sylvain Chavanel and could not take part in the final sprint.
The peloton went through Sorbiers, where the late Kazakh rider Andrey Kivilev was buried after he died following a crash in March 2003 during the Paris-Nice race.
His death prompted the International Cycling Union (UCI) to implement the mandatory use of helmets in all races.
It brought back memories to fellow Kazakh Alexandre Vinokourov, now the Astana manager who was Kivilev's best friend.
"It's a bad memory, but there are also good memories. I remember we arrived in France, in St Etienne, together in 1997, we signed our first contracts together," Vinokourov told Reuters before the start of the stage in Bourg en Bresse.
"Every year I go to Nice to see his son, Leonard."
Europcar sports director Andy Flickinger, who was a rider at the time, was emotional.
"I was in the race when it happened," he told Reuters.
"Kivilev was a good guy, one of the first from his country coming to Europe. At the time I would not wear a helmet, now everybody does even if I'm not sure there are more accidents. But it's good that riders wear them."
Friday's 13th stage takes the peloton to the Alps with a mountaintop finish in Chamrousse.
(Reporting by Julien Pretot; Editing by Rex Gowar)
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