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About halfway up Manse, Contador attacked, but Richie Porte responded and got Sky teammate Froome back on Contador's wheel.
Contador attacked again soon after and opened up a gap of about 50 meters (yards) before being caught, and then tried a third attack.
Bauke Mollema could have gained more time but chose instead to wait for Froome and Contador after their mishap. Mollema remains second overall, 4:14 behind Froome, while Contador remains 4:25 back in third.
Froome will look to extend that lead in Wednesday's time-trial — one of his favored disciplines — before three huge mountain climbs in the Alps.
"It's a very hard day tomorrow with two climbs and two downhills," Froome said. "You have to keep a bit of energy for the following days in the Alps."
The 168-kilometer (104-mile) route started from the Provence village of Vaison-la-Romaine in the heart of the Rhone winemaking country. Early on, about 25 riders started to pull away but started to split open when Costa surged up Manse.
"This win will make us happy," said Costa, whose teammate Alejandro Valverde dropped out of podium contention last week when he lost a lot of time on Friday's 13th stage after stopping for a mechanical failure. "It's my victory, but it's also a victory for the team."