Rogers gave his Tinkoff-Saxo team their second stage win this year by powering away from a breakaway group in the 237.5-km mountain trek from Carcassonne.
Vincenzo Nibali of Italy retained the overall leader's yellow jersey ahead of Spain's Alejandro Valverde while France's Thibaut Pinot, who was the strongest in the intimidating ascent to the Port de Bales, moved up to third at the expense of compatriot Bardet.
AG2R-La Mondiale rider Bardet, who had promised to blow the race open in the Pyrenees, was unable to sustain the pace in the climb and lost over 1:40.
Van Garderen could not follow either and lost more than three minutes, but Bardet's team mate Jean-Christophe Peraud could not be shaken off.
World champion Rui Costa of Portugal did not start the stage as he withdrew suffering from pneumonia.
"He was forced to withdraw after an X-ray examination, carried out during the rest day at a clinic in Pezenas, showed an outbreak of pneumonia," his Lampre team said in a statement.
A 21-man breakaway took shape after about 60 kilometres and built a 12:30 gap that started to shrink when they hit the first slopes of the ascent to the Port de Bales, a 11.7-km climb at an average gradient of 7.7 per cent.
Astana and Movistar set the tempo of the peloton in the climb while Rogers's acceleration in the break reduced the group to seven men.
Thomas Voeckler's attack skimmed it down to four before his Europcar team mate Cyril Gauthier burst away, but he quickly paid for it and was dropped.
Rogers, Colombian Jose Serpa and Voeckler were on their own three kilometres from the summit.
That is where Pinot's team mate Arnold Jeannesson upped the pace in front of the yellow jersey group as Bardet dropped out. Van Garderen was long gone already.
Pinot finished it off in the last kilometre, dropping Valverde, Peraud and even Nibali.
However, they all came back in the descent and worked together with several team mates who had been dropped from the morning's breakaway and they all finished together.
(Editing by Ed Osmond)
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