The 188km stage from Pau, which included the legendary Col d’Aspin and Tourmalet climbs, proved ideal for a breakaway to survive. Majka took advantage of that possibility to finish one minute ahead of a charging Dan Martin (Cannondale-Garmin) and German road race champion Emanuel Buchmann (Bora-Argon 18).
The top of the general classification remained unchanged, with Chris Froome (team Sky) leading Tejay van Garderen (BMC) and Nairo Quintana (Movistar) by the time gaps created on Stage 10.
However, defending champion Vincenzo Nibali (Astana) again struggled in the finale to Cauterets-Cambasque, losing more time to Froome and dropping out of the general classification's top 10.
The pace was high in the early part of the stage, with a series of attacks in the first 80km. The peloton eventually settled on Majka, Buchmann, Thomas Voeckler (Europcar), Serge Pauwels (MTN-Qhubeka), Steve Morabito and Arnaud Démare (FDJ) and Julien Simon (Cofidis) as the breakaway selection of the day.
The seven riders hit the Col d’Aspin with a lead of more than six minutes and were later joined by Martin, who led the break across the top of the climb. Majka then turned on the gas at the foot of the Tourmalet and ground away from the break, shredding its cohesion.
The same happened to the yellow jersey group as it was winnowed down to 30 riders by the pace. Quality names were dropped by the pace while others, like former world road champion Rui Costa (Lampre-Merida), saw no future and abandoned.
Majka crossed the summit of the Tourmalet, the highest pass of the 2015 Tour, with five minutes and 35 seconds in hand on the yellow jersey group. By this time the bunch had been reduced to 14 riders, including Nibali, Froome, his team-mates Richie Porte and Geraint Thomas, Alberto Contador (Tinkoff-Saxo), Quintana and his Movistar team-mate Alejandro Valverde, Van Garderen and Samuel Sanchez (BMC), Tony Gallopin (Lotto-Soudal), Pierre Rolland (Europcar), Robert Gesink (Lotto-Jumbo) and Bauke Mollema (Trek Factory Racing).
In the meantime, Pauwels and Martin remained in no man's land between Majka and the Froome group. While Pauwels initially looked like he would hold on for second on the final climb to Cauterets-Cambasque, but a searing surge by Martin in the last few kilometres saw him overtake Pauwels as the MTN-Qhubeka rider faded.
Buchmann too found his second wind as the finish approached and also passed to score the best result of his young career.
The Froome group finished a little more than five minutes later, with Nibali the only absentee. His Tour defence is now truly in tatters.
Majka dedicated the win to team-mates Ivan Basso, who withdrew from the Tour to undergo surgery for testicular cancer, and Daniele Bennati.
"I only attacked once, but it was at the right moment. I was watching the other riders and noticed that many of them were suffering so I decided to attack," said Majka
"A lot of people are asking me whether I want to go for the polka dotted jersey, but it’s not easy to do that and ride for Alberto at the same time."
While the fight for the yellow jersey may have entered a state of detente today, race leader Froome warned that the race is far from over as his rivals are "far from demoralised".
"Maybe for the viewers it's been calm but I can guarantee that a lot of people have left a lot of energy on the road with the temperatures and the high speed of the first two hours of racing," he said.
I can expect to be put in difficulty every day. Tomorrow we'll have a decisive stage finishing at Plateau de Beille and that's a tough climb. A lot of guys will feel the effects of the hard racing we've had today."
Peter Sagan added to Tinkoff-Saxo's successful day by reclaiming the green jersey from Andre Greipel. Sagan now leads the points compeitition by 17 points. Froome remains in the polka-dot jersey.