The three-time Tour winner faded badly on the brutal final climb of the 214.5 kilometre stage from Pau while Aru finished third to take the final bonus seconds on offer.
The Italian national champion now leads the Tour by six seconds from Froome, with Frenchman Bardet closing to within 25 seconds of yellow in third place overall.
Astana’s Aru will wear the yellow jersey for the first time in his career, while Froome has handed it over for the second time in his, having also lost it after stage four in 2015 before reclaiming it three days later and carrying it to Paris.
“It was a tough day for me in the final metres and my team-mates had done such an amazing job,” Froome said.
“But I didn’t have the legs to finish it off, it’s as simple as that. No excuses.”
How he might have wished for the legs he had on these roads in 2012, when he was made to wait for Sir Bradley Wiggins when he wanted to attack in search of a stage win.
But there was a key difference this year, with the stage finishing at the top of the airstrip used in the opening sequence of James Bond film Tomorrow Never Dies, and it was there Froome fell short.
“The finish is brutal,” Froome added. “Ramps of over 20 per cent, that’s a really hard finish and I can only say congratulations to Romain Bardet for winning the stage and to Fabio Aru for taking the yellow jersey. The race is certainly on now.”
Froome came into this race expecting Richie Porte to be the biggest threat, but though the Australian has crashed out, he has found himself in his toughest battle yet.
AG2R La Mondiale’s Bardet timed his attack perfectly, taking victory ahead of Aru and the Colombian Rigoberto Uran of Cannondale-Drapac, both two seconds back but picking up bonus seconds.
Uran sits fourth in the general classification, 55 seconds back after a 20-second penalty for taking an illegal bottle in the last five kilometres.
It was a third career stage win for Aru, who finished second overall last year.
Froome crossed the line in seventh place.