New Zealand's Bauer was part of a two-man break with Swiss national champion Martin Elmiger (IAM Cycling) that was thwarted right at the death as the peloton left it late on a day of heavy showers and thunderstorms.
Milan-San Remo winner Kristoff powered past Elmiger and Bauer on the closing straight to take his second win of the Tour ahead of Australia's Heinrich Haussler (Garmin-Sharp) and Slovakian Peter Sagan (Cannondale).
German national champion Andre Greipel (Lotto-Belisol) and Australia's Mark Renshaw (Omega Pharma-Quick Step) completed the top ten as a broken Bauer came home in tenth place after coming so close to tasting victory.
"It was close and we left it really late but we managed to catch them on the line and I'm really happy," said Kristoff, winner of Thursday's stage 12 at Saint-Etienne.
Italy's Vincenzo Nibali (Astana) survived a little scare in the crosswinds to finish safely in the main pack and retain his comfortable lead in the overall standings going into the race's second rest day at Carcassonne.
"It was a pretty quiet day for the most part," said Nibali, who has a 4:37 lead over Spain's Alejandro Valverde (Movistar) and another 13 seconds on third-place Romain Bardet (Ag2R-La Mondiale).
"There was a lot of wind which was tricky and dangerous - and in the final part, with the roundabouts and rain, was pretty challenging. But we survived."
TWO-MAN BREAK: Bauer and Elmiger broke clear just two kilometres into the long transitional stage through the lavender fields of Provence and Vaucluse - meaning they clocked up a total of 210km on the front of the race.
The pair built up a maximum lead of over seven minutes in cooler temperatures but bright sun as the race bade adieu to the foothills of the Alps. Reports of stormy weather at the finish at Nîmes reached the earpieces of the riders in the peloton and the pace.
Fierce wind with around 70km remaining saw the peloton string out and caused numerous splits, with the polka dot jersey Joaquim Rodriguez (Katusha) and Australia's Richie Porte (Team Sky) among a dozen or so riders caught out.
The BMC team of fifth place American Tejay Van Garderen took over the pace-setting from Bardet's Ag2R team and Nibali, the yellow jersey, was forced to ride up through the peloton to keep safe as the peloton sped towards the storm clouds.
Rain started to fall just ahead of the intermediate sprint at La Galine around 48km from the finish. The torrential showers made the roads slippery and played into the hands of the two escapees, whose lead by now had been whittled down to just two minutes.
With around 30 roundabouts in the final 45km of the race - many of which reduced to potential death traps with the standing water and low visibility - played into the hands of Bauer and Elmiger, who held one minute with 15km remaining. Attempts to bridge the gap by both Michal Kwiatkowski and Tony Martin of Omega Pharma-Quick Step were thwarted as the peloton slowly cut the advantage of the defiant escapees
Once the rain stopped and the sun returned it looked like it would be business as usual for the teams of the main sprinters. But the leading duo dug deep still held 25 seconds going into the final two kilometres and 12 seconds under the flamme rouge. A moment's hesitation probably did for their chances, Bauer eventually leading out the sprint before being caught inside the final hundred metres.
"It's a childhood dream - a fantasy - to try and win a stage on the Tour de France for any cyclist - especially a Kiwi cyclist because there aren't many of us," Bauer said after he regained his composure following teary scenes at the finish line.
"As you can see from my meltdown at the finish, I was pretty disappointed. That's cycling for you: so many highs, so many lows. We both had decent legs in the final twenty kilometres but perhaps I left it a bit too late."
If not matching Bauer's agony, the green jersey Sagan was once again reduced to picking up the crumbs as his search for that elusive stage win continued. "It's my first third place in the race," said Sagan, four times second place this year, in a bid to put a positive spin on things. "I was too far behind and it was too late."
RIDE OF THE DAY: Escapees Bauer and Elmiger share this award for their titanic effort which survived a storm and more than 220km before being sunk in the closing straight.
DAY TO FORGET: Escapees Bauer and Elmiger won't want to remember those final one hundred metres...
COMING UP: The second rest day in Carcassonne is followed by the first of three successive days in the Pyrenees: Tuesday's 237.5km stage includes three lower category climbs ahead of the HC ascent of Port de Bales before downhill ride to Bagneres-de-Luchon that will have yellow jersey Vincenzo Nibali salivating. Perhaps Peter Sagan will go in an early break and downhill to glory?
PLAT DU JOUR: Carcassonne is the cassoulet capital of France so use the rest day to stock up on confit duck legs, sausages and pig's cheek all slow-cooked in a terracotta bowl of white beans.
STAGE IN A SENTENCE: Jack Bauer's mission is terminated by Norwegian hitman in Nîmes.
- Sports & Recreation
- Vincenzo Nibali
- Jack Bauer
- Omega Pharma-Quick Step