Majka was the only rider of a large breakaway to hold off the habitual charge of Nibali, the 24-year-old Polish climber crossing the summit in Risoul 24 seconds ahead of the increasingly bullish Sicilian.
Frenchmen Jean-Christophe Peraud (Ag2R-La Mondiale), Thibaut Pinot (FDJ) and Romain Bardet (Ag2R) competed the top five on the 177km stage from Grenoble, which featured the gruelling ascents of the Col du Lautaret and the Col d'Izoard ahead of a first ever summit finish in the ski resort of Risoul.
Nibali extended his lead in the overall standings to 4:37 over the Spain's Alejandro Valverde, who faded on the final climb to concede exactly a minute to the race leader. Movistar's Valverde will now be looking over his shoulder at youngster Bardet, the white jersey, who trails the Spaniard by just 13 seconds on GC.
Majka's brave and defying win came after a late call-up to Tinkoff-Saxo's Tour squad following the shock suspension of Roman Kreuziger prior to the race. With team leader Alberto Contador crashing out of the race with a fractured tibula last week, Majka entered the Alps with renewed vim and vigour.
Seventh in the Giro d'Italia earlier in the season, Majka was part of the main break in Friday's opening stage in the Alps before forcing himself into Saturday's select 17-man break alongside team-mate Nicolas Roche.
As Roche and his fellow escapees faded on the decisive ascent, Majka rode clear to hold off yet another stinging attack by Nibali back in the group of the main favourites.
"This is the first victory of my career," an ecstatic Majka told Eurosport after his coming-of-age win. "I'm always second or third and so I knew I needed to attack on the final climb. After the Giro I needed to rest but I feel that my form is coming up now."
If Nibali's grip on the yellow jersey looks increasingly unlikely to be loosened between here and Paris, the battle for the remaining podium positions is as hot as the sweltering temperatures that have engulfed the peloton since the first rest day.
French youngsters Pinot and Bardet finished neck-and-neck 50 seconds down on Nibali to pile the pressure on veteran Valverde, who struggled to match their accelerations on the final climb. Bardet, in particular, benefited from some excellent team-work as Ag2R-La Mondiale increased the tempo on the descent of the Izoard to momentarily distance the likes of Pinot and the American Tejay Van Garderen (BMC).
"My team did an incredible job and we took every opportunity today," said Bardet. "In the end, I took time back on all my rivals for the top ten."
Bardet trails 29-year-old Nibali by 4:50 on GC with second-place Valverde just 13 seconds clear. FDJ's Pinot - his compatriot and rival for both the podium and white jersey - is a further 16 seconds adrift in fourth place, while Van Garderen - sixth in Risoul - completes the top five at 5:49.
SEVENTEEN-MAN GROUP: Tinkoff-Saxo were not the only team needing to rethink their tactics following the withdrawal of their top rider - and Team Sky threw two men in an early break after seeing their GC hopes go up in flames following the withdrawal of Chris Froome and the collapse of Richie Porte.
The presence of Geraint Thomas and Mikel Nieve for Sky counteracted Tinkoff's duo of Majka and Roche in a large break that also included the green jersey Peter Sagan and his Cannondale team-mate Alessandro De Marchi, Spanish climber Joaquim Rodriguez (Katusha), moustachioed Colombian Jose Serpa (Lampre-Merida) and Britain's Simon Yates (Orica-GreenEdge), the youngest rider on the race.
Combining well, the break built up a maximum lead of around five minutes as Sagan picked up maximum points in Bourg d'Oisans at the intermediate sprint. Resisting the temptation of taking the turn to Alpe d'Huez, the riders embarked on the long 32km climb of the Col du Lautaret, with Rodriguez crossing the summit in pole position to return to the top of the polka dot jersey standings.
The gap came down to just three minutes as the riders passed through the feed zone at Briancon ahead of the highest climb of the race, the 2,360m Col d'Izoard. Sagan was first to fade, the Slovakian's thoughts perhaps moving on to Sunday's stage to Nimes where he will try and snare that elusive win after four runners-up spots to date.
Thomas drove the pace of the break for most of the climb, seemingly burying himself (as is his wont) for Sky team-mate Nieve. Rodriguez crossed the summit in first place to trouser more KOM points and secure the Souvenir Henri Desgranges prize - but the gap had been whittled down to just two minutes as the riders passed through the famous Casse Deserte - Broken Desert - of the southern side of the Izoard.
Back with the main pack, Bardet piled on the pressure with the help of his Ag2R team-mates to distance the yellow jersey and the likes of Pinot, Valverde and Van Garderen. But it has all come back together by the time the main pack started the final ascent - making its first appearance on the Tour.
De Marchi - winner in Risoul during the 2013 Criterium du Dauphine - attacked from the breakaway, but was soon caught and passed by Majka. In pursuit Rodriguez and Serpa caught the Italian before Serpa faded. Ag2R set a blistering pace up the climb to swallow up the remnants of the break.
After a couple of half-hearted attempts by Frenchman Pierre Rolland (Europcar), Nibali made his move with 3.5km remaining. French veteran Peraud was the only rider who could match his unseated surge, the pair quickly catching Rodriguez and De Marchi as they bore down on Majka.
Nibali made numerous unsuccessful attempts to distance the sandbagging Peraud and Majka still had 30 seconds to play with going under the flamme rouge. Content to take more time off his rivals rather than chase a fourth stage win, Nibali eased up and Makja secured only a second ever stage win for a Polish rider on the Tour.
A late dig by Nibali saw him cross the line two seconds quicker than 37-year-old Peraud, before the other main favourites jostled for position behind, in the knowledge that Valverde was having an off day.
RIDE OF THE DAY: If Rafal Majka was livid when he learnt of his late selection in Tinkoff-Saxo's Tour de France squad, it's fair to say that the Pole will now be happy he's topping podiums rather than topping up his tan this July.
DAY TO FORGET: Arnaud Demare was the latest rider to suffer acute illness in the saddle, the French national champion having to answer a call of nature in a road-side camver van on the ascent of the Lautaret. Alejandro Valverde also had a stinker, the Spaniard losing one minute to his podium rivals Romain Bardet and Thibaut Pinot to see his second place on GC under threat.
COMING UP: Sunday's 222km stage 15 from Tallard to Nimes is a rolling affair but features no categorised climbs before a flat finish. If this is a rest day for the men on GC, you can get your bottom dollar that Cannondale will move heaven and earth to finally deliver their man in green, Peter Sagan, to the finish line ahead of everyone else - and not behind someone else (as has been the custom on four occasions to date)
- Sports & Recreation
- Romain Bardet
- Thibaut Pinot
- Jean-Christophe Peraud
- Joaquim Rodriguez
- Alejandro Valverde
- Van Garderen