The 28-year-old extended his overall lead in the general classification to four minutes 14 seconds over Bauke Mollema with victory in stage 15, which also saw Alberto Contador crack seven kilometres from the summit.
Froome kicked at a decisive moment to leave Contador floundering as he attempted to reel in Nairo Quintana, who was out in front on his own.
After catching the Colombian, Froome powered to a 29-second victory to tighten his grip on the yellow jersey.
"My main objective was to get more of a buffer on the general classification riders but I didn't see myself winning the stage," Froome told ITV Sport.
"I wasn't expecting it to be that hard to catch Quintana and when I did I thought he was going to win the stage, but in the last two kilometres he started fading - I didn't really attack, he just couldn't hold my wheel.
"This climb is so historic and means so much to this race, especially in the 100th edition of the race. I really can't believe it."
The riders have a rest day before Tuesday's 168km ride from Vaison-la-Romaine to Gap.