Most of the main contenders were in the second group, 73 seconds further back.
Colombian Rigoberto Uran took second place for six bonus seconds while Froome pipped Frenchman Romain Bardet to the line for third.
That sees three-time Tour winner Froome extend his lead, with Uran and Bardet both now 27 seconds back.
Italian national champion Fabio Aru was dropped on the Galibier and conceded more than 30 seconds to lose his grip on second place overall.
Aru dropped to fourth, 53 seconds off yellow. Froome’s team-mate Landa sits fifth . Irishman Dan Martin moved up one to sixth with a battling performance 24 hours after he was caught out in crosswinds and dropped two places.
For Team Sky’s Froome it was a case of job done as he ticked off the first of two Alpine stages which could decide this year’s Tour even before Saturday’s time trial in Marseille.
“I didn’t have the legs in the Pyrenees,” said Froome, who had lost yellow on stage 12 to Peyragudes before reclaiming it 48 hours later.
“But now I feel strong. I’ve also got to look towards tomorrow because it’s a summit finish on Izoard and I’ll have to see how the legs are.”
The main rivals all tested each other on the long climb of the famed Galibier, the highest point on this year’s Tour with a summit at 2,642m, but they could only shake off Aru.
Simon Yates, wearing the best young riders’ white jersey, also struggled, finishing three minutes and 14 seconds behind Roglic. He conceded 90 seconds to rival Louis Meintjes in the battle for white, though Yates still has a cushion of two minutes 28 seconds.
LottoNL-Jumbo’s Roglic emerged from a sizeable breakaway, partly powered by veteran Alberto Contador.
Roglic, pictured below, attacked solo 6km from the summit of the Galibier, the second of two hors categorie climbs, and made it pay to deliver Slovenia’s first Tour de France stage win.
The 27-year-old, who gave up ski jumping in 2011 and began professional cycling two years later, said: “This is the biggest event in cycling. I am here after only five years on my bike. Winning is unbelievable.”
Only 29 seconds had separated the top four on the general classification at the start of the day and they closely marked each other all the way.
Martin was the first to attack before AG2R La Mondiale’s Bardet put in a dig. Aru was struggling and they took it in turns to ensure the Astana man was left behind.
Froome still had team-mate Landa with him as they tried to set up the sprint finish for the bonus points, but although the Briton made it round Bardet he could not beat Cannondale-Drapac’s Uran to the line.
Early in the stage Marcel Kittel, wearing the points leaders’ green jersey and winner of five stages in this Tour, had to abandon after a crash.
The German had hit the deck in an accident that also brought down Warren Barguil in the polka dot jersey and British national champion Steve Cummings, who both recovered.
Kittel began the day with a 29-point lead over Australian Michael Matthews, but had already seen that cut to nine as his Team Sunweb rival won the day’s intermediate sprint.
Kittel was not alone in abandoning on one of the Tour’s toughest days. Frenchman Thibaut Pinot, a contender in previous years, also quit, as did British sprinter Dan McLay of Fortuneo-Oscaro.