The Irishman was second over the line on the mountaintop finish on stage five, 16 seconds behind the stage winner Fabio Aru (Astana).
Aru jumped clear of a select front group of riders who came into the base of the category one-ranked ascent together but as the road rose skyward that cluster of top contenders became noticeably trimmed back.
Martin, a former stage winner, looked to be in real difficulty as Froome and overnight leader Geraint Thomas’s Team Sky riders forced a ferocious speed.
An acceleration by Froome put his teammate Thomas and a host of others out the back and inside a few kilometres to go it was really down to Martin, Richie Porte (BMC Racing), Simon Yates (Orica-SCOTT), Alberto Contador (Trek Segafredo), Romain Bardet (Ag2r La Mondiale) and Rigoberto Uran (Cannondale Drapac).
Aru went for broke with a trademark acceleration at the steepest part of the climb and the Italian national champion powered clear, never to be seen again before the finish line.
Behind him, the group was down to five with Martin clinging to that knot of rail-thin climbers.
And incredibly, just when it seemed like he’d emptied his tank he attacked and forced a gap that nobody could — or was willing to — close.
He was just 16 seconds down on Aru when he fell over the line but he took a very precious four seconds back on Froome, adding to the six he took on his rivals on Monday.
Froome and Porte were third and fourth on the stage, four seconds down, while a further four seconds down came Bardet and two more seconds back the road came another big trio.
Yates, Uran and Contador (Trek-Segafredo) filled places sixth through eighth, 10 seconds down on Martin.
Geraint Thomas relinquished the race lead as he was 40 seconds down on Aru and 24 down on Martin, though the Welshman is still in second at 12 seconds.
The only other Irish rider in the race was Nicolas Roche and he performed heroically in the service of Richie Porte, finishing 13th just 1:05 down on Aru and he is 21st now on GC at 2:14.
Meanwhile, world champion Peter Sagan reacted to his expulsion by saying he believes he did nothing wrong that led to the crash which saw Mark Cavendish break his scapula and subsequently leave the race yesterday.
“What can I do?
“I can just accept the decision of the jury but for sure I don’t agree with them because I think I didn’t do something wrong in the sprint,” he said before the stage start.
“What is bad is that Mark fell down and it’s important that he can recover well. I’m sorry for that.
“As you see can see on the internet, it was a crazy sprint finish .
“It is not the first one like that and won’t be the last one like that. So I wish to Mark to recover well and that’s it.”
Cavendish confessed the relationship with Sagan will remain good but he commended the race jury who booted him out.
“I think in my career there is always going to be an opinion on me – I have a lot of good fans but I have a lot of people who don’t like me,” he said.
“There’s always going to be opinion on Peter, he’s a great person for the sport. I think what you have to do here is take away the riders involved, take away the jerseys involved and look at what happened. And that’s why we have a jury to make decisions.
“If I’m honest it takes a lot of courage, a lot of balls to eliminate the world champion from the Tour de France, and I commend the jury on taking a decision that wasn’t based on influences from social media or outside.”
The race continues today with a 216-kilometre journey from Vesoul to Troyes and though there are two category four climbs a sprint is the most likely outcome.
Martin will be hoping he gets through the day okay and avoid any major time losses before two mammoth stages on Saturday and Sunday.