Wiggins received the leader's yellow jersey from Fabian Cancellara following the 199-kilometer trek from Tomblaine up to the Vosges mountains ski station of La Planche des Belles Filles. The Swiss time-trial specialist had led since winning the opening prologue a week ago.
Wiggins, who began the day seven seconds behind Cancellara in second place, leads defending champion Cadel Evans of Australia by 10 seconds. Vincenzo Nibali of Italy is also 10 seconds off the pace in third. Cancellara trailed by 1 minute, 52 minutes.
"It's a great day for the team, we won the stage and took the yellow jersey," Wiggins said. "This is my first time in the yellow jersey. It's incredible — it's been a dream of mine since I was a kid."
Wiggins, a three-time Olympic track gold medalist, is bidding to become the first Briton to win the Tour and is the first from his country to wear the yellow jersey since David Millar in 2000.
With two time trials and climbing days in the Alps and Pyrenees still to come, Wiggins disagreed he had taken the lead too early with the finish in Paris on July 22.
"You can't get too cocky in this race and choose when you take the yellow jersey. I'd much rather be in yellow than in hospital, like half the peloton," Wiggins said, referring to crashes on Friday that forced at least 12 riders to quit the race.
Wiggins crashed out of the 2011 Tour because of a broken collarbone and said he felt "lucky" he has been trouble-free this year.
Froome was part of the Team Sky phalanx that powered up the final climb. With most rivals falling away, the Kenyan-born Briton burst ahead to finish two seconds in front of BMC leader Evans and Sky leader Wiggins.
Froome, who took the polka-dot jersey as the Tour's best climber, said he was surprised Evans couldn't keep pace. The Australian was puffing and his face glistening with sweat as he crossed the finish a split-second before Wiggins.
Wiggins said he is focusing on Evans as his biggest rival for the title.
"You saw it today, Cadel never gives up," Wiggins said.
The stage marked the first of three summit finishes this year. Lighter, nimbler mountain specialists seized the limelight after a first week dominated by sprinters across the flatter regions of Belgium — where the race began June 30 — and northern France.
The final ascent, at 5.9 kilometers, was relatively short as far as the Tour's biggest climbs go. But it was steep, with a grueling 14-percent gradient in the last 500 meters.
Seven riders broke away from the pack after 15 kilometers and held the lead until the start of the final climb. Wiggins' Team Sky then pressed the peloton's pace before overtaking.
Giro d'Italia champion Ryder Hesjedal of Canada pulled out before the start of the stage after being injured in a multi-rider crash Friday.
The three-week race takes riders on another bumpy ride Sunday, with seven climbs in the 157.5-kilometer course from Belfort to Porrentruy.
