Helped into the day's break by team mate Daniel Oss, Kämna spent some 140kms at the front of the race, attacking former breakaway companion Richard Carapaz (INEOS Grenadiers) 300 metres from the top of the category 1 Montée Saint Nizier du Moucherotte.
Despite a minute in hand over Carapaz as he hit the final climb, the German kept his foot on the gas, powering up the category 3 climb to the finish, the effort etched on his face.
He wasn't going about to let this victory slip through his fingers after Daniel Martínez (EF Pro Cycling) had his measure with a last minute kick at the line on Stage 13.
It is Kämna's first grand tour stage victory and only his second career win after claiming Stage 4 of the Critérium du Dauphiné.
"I'm feeling great," he said. "Absolutely awesome day for me now. It was a fight from the beginning on.
"I knew I had to make it to the finish alone and when I saw that Carapaz was dropping the speed I thought OK now it's the moment to go and then I just went all in until the end.
"It's a big big big relief also for the team and for me also."
"I can almost not imagine it, the step I made this year is like huge and I'm so blessed to win today."
The yellow jersey battle was mellow as all of the favourites came home together although Miguel Ángel López expended a lot of effort on the approach to the line with not even one second improvement to his overall standing.
In the King of the Mountains classification Benoît Cosnefroy still wears the dots but Pierre Rolland is on equal points after he cleaned up on the two category 2 climbs on today's stage.
The day opened with an intense stanza as various riders eagerly sought their team's first stage win and personal glory. Nans Peters (Ag2R) was one who sparked early attacks including his ascent first over the category 4 Côte de Virieu at 12.5kms.
With the intermediate sprint tantalisingly making an appearance at 44kms before the first category 2 climb, Peter Sagan (BORA-hansgrohe) desparately tried to join some other early combatants but Sam Bennett and his Deceuninck-Quick Step team immediately shut him down.
After 22 kilometres Kämna, Oss, Carapaz, Andrey Amador (Ineos Grenadiers), Julian Alaphilippe (Deceuninck-Quick Step), Sébastien Reichenbach (Groupama-FDJ), Alberto Bettiol (EF Education First), Winner Anacona, Warren Barguil (Arkéa-Samsic), Imanol Erviti, Carlos Verona (Movistar Team), Matteo Trentin (CCC Team), Chris Juul Jensen (Mitchelton-Scott), Nicholas Roche (Team Sunweb), Quentin Pacher (B&B Hotels-Vital Concept) pulled clear, establishing a minute's advantage by the 40th kilometre.
After 41kms raced, Rolland left the peloton in pursuit of the leaders. Team Sunweb unhappy with just Roche in the break, sent Tiesj Benoot and Casper Pedersen up the road. Behind them by about a minute were Sivakov, Powless, Geschke, Nieve and Sicard.
Trentin took the maximum intermediate sprint points over Daniel Oss and the break took care of the rest.
The peloton then relaxed a little, the leaders' advantage increasing to six minutes by the time the race hit the Col de Porte. By the finish, it would reach a maximum back to the main bunch by over 16 minutes.
After claiming the points on the Col de Porte and the Cote de Revel, Rolland waited for the rest of the chasers. His team mate Pacher soon after with 35 kilometres to go and hit the category 1 Montée Saint Nizier du Moucherotte iwth a lead of 30 seconds over his former breakaway companions.
Alaphilippe, Carapaz, Kämna and Reichenbach joined Pacher four and a half kilometres from top but Pacher lost touch one kilometre later.
Carapaz jumped 2kms from the top ultimately dropping Alaphilippe and his second explosion saw Reichenbach fall away. But Kämna stuck to Carapaz' wheel until he attacked 300m from the top of the summit, quickly opening a significant gap on the descent.
The 2020 Tour de France continues with the brutal 170km stage from Grenoble to Méribel (Col de La Loze) - the highest point of this year's Tour is on the menu. Stage 17 begins via SBS On Demand and the SBS ŠKODA Tour Tracker from 8:05pm (AEST) on Wednesday, with the television broadcast set to start at 9:35pm on SBS.