Froome entered the Tour as the favorite after finishing second last year behind countryman and Sky teammate Bradley Wiggins, who isn't defending his title because of an injury. After Froome's performance, the race looks like it's his to lose.
"I must be among the happiest men in the world today," Froome said. "There's a long way to go until Paris. There are two weeks left but we want to keep the yellow jersey."
Froome leads Contador by nearly two minutes, with former champions Andy Schleck and Cadel Evans much further back, ahead of another tough mountain stage on Sunday.
"More than anything today we've got a bit of a psychological advantage over the others," Froome said. "It's quite hard to think about this, standing in yellow today. This is incredible. We've worked for months to be in this position."
In the overall standings, Froome is 51 seconds ahead of teammate Richie Porte and leads third-place Alejandro Valverde by 1:25. Meanwhile, Contador is 1:51 behind in seventh spot; Schleck is 4:00 back in 21st and Evans is 4:36 adrift in 23rd.
This is the 100th edition of the Tour—and the first since Lance Armstrong was stripped of his seven straight titles (1999-2005) for doping.
Froome was asked after the stage if he is riding clean.
"One hundred percent," he responded.
"It's normal that people ask questions in cycling," he said. "I certainly know the results I'm getting, they're not going to be stripped ... I think the sport (has) changed. If you look at it logically the sport is in a better place now than it has been."
The 28-year-old Froome attacked early into the day's second big climb up to Ax 3 Domaines and only Porte, who finished the stage 51 seconds behind in second, was anywhere near him.
"The team was absolutely perfect today," Porte said. "I'm absolutely finished but it was an incredible day."
Contador grimly held on as long as he could, and was dropped by Froome long after 2010 champion Schleck and the 2011 champion Evans had already been left behind.
"I tried to find my rhythm because it was a bit too fast for me," Schleck said. "It's only the first day of the mountains and it's not over yet."
Froome took the yellow jersey from South African cyclist Daryl Impey, his former training partner, who crawled over the line 7:50 behind in 35th spot.
Contador finished the stage 1:45 behind Froome, Schleck trailed by 3:34 and the 36-year-old Evans was 4:13 adrift.
"I wasn't expecting that much for today," Froome said. "The time gaps were so big, that's quite something. We're in a really good position."
Given that Froome may attack again in Sunday's second tortuous Pyrenean climbs—featuring four straight category 1 ascents—he could be well on the way to victory by Monday's rest day.
"I think we're well poised," Froome said. "We're going to have to fight for it but I'm confident in the team we have."
Saturday's 195-kilometer (121-mile) trek started from Castres and stayed flat for a long time leading to the steep climbs.
The Col de Pailheres came first—a ferociously tough ascent for about nine miles at a gradient of eight percent—and then a shorter but even steeper ride to the finish at the ski resort of Ax 3 Domaines.
Colombian rider Nairo Quintana broke away to launch a brutal attack up Pailheres, and only Frenchman Pierre Roland initially followed him, but Porte helped Froome steadily gain ground on them.
Schleck, Evans and then Rolland were dropped with about 6 kilometers (4 miles) left to go up to Ax 3 Domaines as Contador desperately tried to stay on Porte's wheel.
Porte had tired Contador to the point that the Spaniard was even struggling to keep up with teammate Roman Kreuziger, so Froome decided it was time to finish him off.
"I am little bit surprised," Froome said. "Maybe he (Contador) didn't have his best day today."
Froome climbed with such blistering acceleration it almost looked like he was riding a time trial, blowing past Quintana and then Porte.
"It may look easy on TV but it wasn't," Froome said.
Still, he had time to ease up in the saddle, raise his arms and soak up victory in the blazing sun.