Dutchman Bauke Mollema, the strongest and smartest of the breakaway riders, won the 189.5km stage from Laissac-Severac L’Eglise but the action was elsewhere.
Defending champion Froome was first caught at the end of a split caused by French team AG2R-La Mondiale, who continued to press the pedals when the race leader suffered a mechanical problem shortly before the key climb of the day.
The Team Sky leader kept his cool, however, and powered his way back to stay on top of the general classification with an 18-second advantage over Italian Fabio Aru.
Bardet, who enjoyed huge crowd support in his home region in the Massif Central, stays third 23 seconds off the pace. Colombian Rigoberto Uran is fourth six seconds further back.
A week after Aru had attacked just as he suffered a mechanical problem, Briton Froome again had trouble with his bike and, as the race was on, none of the main contenders waited for him.
AG2R-La Mondiale accelerated suddenly at the front of the peloton with some 45km left and Froome and some of his teammates were trapped at the wrong end of the split.
Froome suffered a mechanical problem and was assisted by team mate Michal Kwiatkowski.
Mikel Nieve and Vasil Kiryienka pulled Froome as they trailed Bardet’s group, which contained Aru and Rigoberto Uran, by 45 seconds.
The Briton made up the time on the ascent of the Col de Peyra Taillade, an 8.3km climb at an average gradient of 7.4 per cent.
Bardet and Aru jumped away on the climb but were easily reined in.
Ireland’s Dan Martin pedalled away on a flat section after the descent to gain 12 seconds on Froome and the other favourites.
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