The Belgian BMC Racing rider powered away from a small group on a steep final straight to outsprint Slovakian Peter Sagan, who is himself developing a reputation for failing at the finish.
Van Avermaet, a specialist at one-day races, was third in the Tour of Flanders and Paris-Roubaix this year, having also collected scores of top-five finishes in the classics yet very few big wins in his career.
He was the strongest on Friday, however, as Sagan endured his 15th second-place finish on a Tour de France stage.
"It's a superb Tour for us," said Van Avermaet, who secured BMC Racing their third win on this year's Tour.
"I know I'm not the guy who wins 20 races every season," he added. His previous stage win on a grand tour was on the 2008 Tour of Spain.
Sagan, who retained the green jersey for the points classification, said: "I had nothing left in the last 100 metres.
"(It was) My stupid mistake. After I got his wheel, I sat down."
It was Sagan's fourth second-place in this year's Tour de France.
Belgian Jan Bakelants took third place for AG2r-La Mondiale as Chris Froome retained the overall leader's yellow jersey by finishing safely with the other top riders, seven seconds off the pace.
Team Sky's Froome leads American Tejay van Garderen (BMC Racing) by 2:52 and Nairo Quintana (Movistar) of Colombia by 3:09.
A six-man breakaway built a four-minute gap in searing heat as temperatures reached 40 Celsius in southwest France but they were reined in near the end, with the leading three being caught inside the final kilometre.
Van Avermaet started his effort about 300 metres from the line and held off Sagan with Bakelants finishing three seconds behind.
Frenchman Jean-Christophe Peraud (AG2r-La Mondiale), second overall in last year's Tour, crashed heavily 134 km into the 198.5-km ride from Muret but he got back on his bike despite sustaining deep bruises on his left arm and leg.
Peraud had already dropped out of contention in this Tour after failing to stay with the leading riders in the Pyrenees.
Saturday's 14th stage is a 178.5-km trek between Rodez and Mende, ending with a short yet demanding climb.
"There will definitely be some action between the GC (general classification) guys," said Froome.
(Reporting by Julien Pretot; editing by Toby Davis)
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