Thibaut Pinot, however, kept pace with Nibali and replaced Bardet as France's top podium hopeful: He rose to third.
For the second straight year, the entree to the Pyrenees has dented Van Garderen's podium ambitions.
"I had high hopes for a podium and now it looks like it's taken a big hit ... I just didn't have the legs," said the American.
A year ago, he lost more than 10 minutes to the main contenders, including Chris Froome who went on to win, as they rode up to the Ax 3 Domaines ski station on Stage 8. The year before, in his Tour debut, he lost seconds in the title quest during two Pyrenean stages, but still finished fifth overall and took home the white jersey given to the best young rider.
For Rogers, the 237.5km leg from Carcassonne to Bagneres-de-Luchon - the longest stage this year - was one of vindication and overdue Tour glory.
"Every cyclist's dream is to win a stage at the Tour de France," said Rogers, who also won two stages on Italy's Giro this year. "I can't describe the joy I felt in the last 500m ..."
Rogers came close to not riding in the Giro or the Tour at all.
In a ruling in April, the International Cycling Union accepted that meat Rogers ate in China last year probably caused his positive doping test at the Japan Cup shortly afterward. He convinced the UCI that he had not intended to cheat.
Rogers was suspended after that positive test. Underscoring the pressure, Cycling Australia said at the time that it would seek a maximum two-year ban if he had been found guilty of doping.
He knew the effects of clenbuterol, which helps to build muscle and burn fat, on riders' careers.
Contador lost his 2010 Tour title and served a two-year ban after testing positive for it in the final week of that race. The Spaniard also argued that he ingested it through food, but lost his case.
Rogers might not have had a chance to win a Tour stage if Contador were still racing this year, because his job would have been more of a support rider. Once the Spaniard crashed out, the team's Plan B was to aim for stage victories.
"I can be grateful, but I'm also very heartbroken that Alberto's not here," Rogers said.
As the day began, a breakaway group of 21 riders came together early and chiselled out a lead of more than 12 minutes.
Their unity disintegrated on the 12km Port de Bales, which is considered so hard that it's beyond classification in cycling's ranking system.
In its steepest patch, the gradient reached 11 per cent. Rogers was in a bunch of five riders that came together in the final descent, and he stepped on the accelerator with less than 5km left.
Nibali now leads second-place Alejandro Valverde of Spain by 4m 37s, and Pinot is 5m 6s back. Frenchman Jean-Christophe Peraud is fourth, at 6m 8s, and Bardet is 6m 40s behind. Van Garderen is 9m 25s adrift.
It was just an appetiser for the Pyrenees, with uphill finishes to follow today and tomorrow.
Aside from the time trial to come, Stage 17 starting in Saint-Gaudens will be the shortest stage of this year's race, at 124.5km.
It features three hard Category 1 climbs, and an ascent to the Saint-Lary Pla d'Adet ski station.
The final big test will come in the 54km time trial on Sunday, which is expected to determine the final result for a largely ceremonial ride for the yellow jersey a day later on the Champs-Elysees in Paris.
Overall standings
1. Vincenzo Nibali (ITA/AST) 73h 5m 19s
2. Alejandro Valverde (ESP/MOV) 4:37
3. Thibaut Pinot (FRA/FDJ) 5:06s
157. Jack Bauer (NZL/GRS) 26:47
