The two stages were expected to shake up the standings, but Vincenzo Nibali was not only still wearing the yellow jersey, he was farther out front.
In a flip of their finishes a day earlier in the race's entree to the Alps when Nibali won, the Pole and the Italian crossed one-two after the 177-kilometre (110-mile) ride over the 2,360-meter (7,742-foot) Izoard pass — the race's highest point — and a final ascent up to Risoul ski station.
Majka's victory was the first on this Tour by his Tinkoff-Saxo Bank team, which lost main leader Alberto Contador when he crashed out injured on Stage 10.
Majka was not a threat to Nibali: He began the day 97 minutes behind the race leader, who has carried out a methodical, chipping-away strategy against his biggest challengers for the yellow jersey.
"I am really very happy," Majka, who was sixth in the Giro d'Italia this year, said of his first professional victory after chucking his stage winner's bouquet to the crowd. He became only the second Polish rider to win a Tour stage, after Zenon Jaskula in 1993. "I am a little tired, but ... I had a calm first week to help Alberto. It broke my heart to see him leave."
Team owner Oleg Tinkov, a Russian businessman, choked up, wiped his nose, and put on sunglasses.
"We lost Alberto, we had to win," he said through a translator on French TV. "Rafal is a marvelous young rider. We will come back to try to win the Tour one day."
Majka said he did not believe speculation that Tinkoff-Saxo Bank selected him among its nine Tour riders only after the team suspended Czech rider Roman Kreuziger. Just days before announcing its Tour roster last month, the team suspended Kreuziger from competing any more this year because of anomalies in his biological passport, which cycling officials use to fight doping. He has denied any wrongdoing.
Majka, who said he was tired after the strong Giro performance, said he was given assurances by team managers that "you'll do the Tour, but you'll take it easy in the first week," he said. "It's also wrong to think that I'm doing the Tour because Roman Kreuziger isn't. Had he been able to start, we would have both been part of the team."
Victor Petri, a team spokesman, confirmed Majka was in contention for a roster spot before the Kreuziger case.
The Pole was out front early in the stage, joining a 17-rider breakaway behind Spain's Joaquim Rodriguez, the Tour's best climber. They cleared the first big climb, the Lauteret pass, with about a five-minute lead. By the top of the Izoard, they had thinned to 10. As the groupetto splintered on the last climb, and Nibali and the peloton closed in, Majka covered the last eight kilometres alone.
The stage didn't shake up the top five standings, but the day's biggest loser was Alejandro Valverde of Spain: The Movistar team leader held on to his second place but lost a minute to Nibali and saw his gap over third-place Romain Bardet of France slip to 13 seconds. Overall, Nibali leads Valverde by 4:37 and Bardet by 4:50. American Tejay van Garderen was fifth, 5:49 back.
Nibali's strong performance makes the Tour from here to the finish in eight days in Paris looking more and more like a race for podium spots below him.
Giuseppe Martinelli, a manager with Nibali's Astana team and an Italian cycling veteran, said Nibali's "big engine" was making the difference: "It's what makes the difference between a very strong rider and a regular rider."
"Preparation, team strategy, that's all good. But it's the engine that makes the difference in times like this," Martinelli added.
Nibai crossed 24 seconds behind Majka, followed by Jean-Christophe Peraud in third, two seconds slower. Bardet and fellow French rider Thibaut Pinot conducted a two-man sprint and crossed another 24 seconds back. Van Garderen was fifth, 54 seconds behind.
"From the team car, I was told: 'If you still have something in the tank, go for it,'" Nibali said. "I was looking at gaining some time over Alejandro Valverde. I heard that he cracked after I left him."
Valverde, speaking from a team car, said "I didn't crack!" and explained that his trouble stemmed mainly from bad co-ordination with Pinot on the final climb and taking leadership of their bunch. As for the overall title chase, Valverde said: "Nibali is the strongest, but we others are neck and neck."
Nibali addressed speculation that he might have known Michele Ferrari, an Italian doctor who was banned by the Italian Cycling Federation in 2002 and by the U.S. Anti-Doping Agency two years ago for serving as a doping consultant on Lance Armstrong's winning teams.
"I've never met him personally. I've been accused in the past to have worked with him," said Nibali, referring to allegations that there were photographs of him together with Ferrari. "Those pictures simply didn't exist."
Sunday's stage offers some relief after the Alps: Stage 15 is a flat 222 kilometres (138 miles) from Tallard to Nimes, before riders take the second rest day.