The 2013 champion produced a brutal acceleration with just over six kilometres left in the final ascent after his Team Sky lieutenants had prevented attacks by setting a devilish pace in the first part of 15.3-km climb.
When Froome attacked in his usual style -- head bent, his legs spinning at a spine-chilling frequency -- American Tejay van Garderen, Spain's Alberto Contador and defending champion Vincenzo Nibali had already been dropped.
Only Colombian Nairo Quintana was still with Froome, who was also alongside team mate Richie Porte of Australia.
Movistar rider Quintana could not follow and he finished the 167-km trek from Tarbes in third place, 1:04 behind Froome.
Porte was second, 59 seconds off the pace.
"It's a great day, we knew the first 10 kilometres of the climb were very hard. The last part was easier so the goal was to go with about six km left -- just before it got easier -- it was the plan this morning," Team Sky manager Dave Brailsford told reporters.
"Chris is more composed than in 2013."
Overall, Froome leads Van Garderen (BMC Racing) by a massive 2:52, with Quintana in third, 3:09 off the pace.
Contador is sixth, 4:04 behind after the Tinkoff-Saxo leader ended up 11th on the day (+2:51) with Astana's Nibali (21st, +4:25) in 10th, 6:57 adrift with six mountain stages left.
"It was a bad day, for some reason I could not breathe, my legs could not spin like I wanted them to," said Contador, whose bid for a rare Giro/Tour double suffered a major blow.
Movistar's high pace at the front trimmed down the peloton in the first part of the final ascent as Froome stayed safe in the small pack.
Among the first top names dropped were American Andrew Talansky, Ireland's Dan Martin and most notably France's Romain Bardet, sixth overall last year.
His AG2r-La Mondiale team mate Jean-Christophe Peraud also lost contact on a terrible Bastille Day for the French.
Nibali lost touch and ended up more than four minutes behind Froome.
Sky then took over at the front and Quintana's team mate Alejandro Valverde attacked twice but the pace set by Geraint Thomas was so high that the Spanish champion could not create a decent gap.
Contador was dropped with just over seven kilometres left. Sensing the Spaniard's weakness, Porte accelerated and the blow was fatal for Van Garderen, who took 10th place 2:30 off the pace.
Froome then started to spin his legs and he never looked back.
(Editing by Ed Osmond)
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