With the day's main climb beginning, Sergio Henao and Mikel Nieve helped Froome as he battled to rejoin the main group, with Landa dropping back to help him complete the task ahead of the top of the descent, with 33 kilometres still to ride.
It looked like a missed opportunity for Froome's rivals as they could not capitalise on the effort he had spent making his way back over some of the steepest sections of the climb - hearing boos from the strongly pro-Bardet crowd as he went.
"That was extremely stressful, panic station", said Froome after the 15th stage that was won by Dutchman Bauke Mollema (Trek-Segafredo).
Others might have crumbled, but Froome showed a rock-solid performance that carried him to three Tour de France titles so far. After Chris Froome lost his overall lead to Fabio Aru, the team has worked hard to get it back for him. "Kwaitkowski gave me his wheel because the team vehicle was far away".
They were joined by a further 18 riders as the race reached the second of four climbs of the day, which came with almost 100km remaining. They've all been around the block a few times, and maybe when they were younger in their earlier 20s, it would've taken some time.
Matthews won comfortably ahead of Olympic champion Greg Van Avermaet in the tough uphill sprint finish to the 181.5km stage from Blagnac to Rodez, with Norway's Edvald Boasson Hagen third. "For four successive days, he has cut away at the time lost last weekend and is now in fifth place, just 1' 12" behind Froome, and 49 seconds behind third-placed Roman Bardet.
After Monday's second rest day and a likely straightforward stage on Tuesday with a bunch sprint finish, Wednesday sees the Tour head into the Alps for the final mountain showdown of this year's race.More news: Jeremy Meeks & Really Rich Girlfriend Spotted Hangin' in Hollywood
"I think you can do it successfully, but not for GC", White said.
George Bennett has held onto 11th place in the Tour de France after a solid ride in a tricky 14th stage.
"This morning he was three minutes back so when someone's further away in the standings they're given more space (to get away)".
Chris Froome suffered his second mechanical problem at a critical point during a stage.
"There is a lot of trust, no doubt, no question", Team Sky manager Dave Brailsford said after the stage when asked about Landa's ambition. But I made it.
The defending champion came seventh, one second behind stage victor Michael Matthews of Australia as Aru, who started the day with the yellow jersey on his shoulders, lost 25 seconds to the Briton in the final effort, a 600-metre climb at an average gradient of 9.6 per cent.