Sagan, the green jersey from Cannondale still looking for his first win on the race, crashed with 2.5km remaining of the rain-soaked 208.5km stage from Maubourguet. The incident disrupted the chase and acted as a lifeline for Garmin-Sharp's Navardauskas.
A textbook day for Garmin saw Dutchman Tom-Jelte Slagter ride in the day's main break to act as a foil for Navardauskas, who launched his decisive attack on the only climb of the day - the Cat.4 Cote de Monbazillac 15km from the finish.
Navardauskas - a late replacement in Garmin's team for Britain's David Millar - survived a series of wet bends on the approach to Bergerac before soloing to victory with seven seconds to spare over a pursuing group of a dozen riders.
"The opportunity came up but I never thought I would do it," said 26-year-old Navardauskas - winner of a stage in last year's Giro d'Italia. "Sebastian [Langeveld] and Jack [Bauer] paced me on the final climb and then there was a great pull by Tom[-Jelte Slagter] over the top. It was all planned. Tom did an amazing job - his strength was incredible."
John Degenkolb (Giant-Shimano) out-sprinted double stage winner Alexander Kristoff (Katusha) for second place before the peloton arrived in dribs and drabs following the fall-out from the Sagan crash, which also took down French youngster Romain Bardet (Ag2R-La Mondiale) and Frank Schleck (Trek Factory Racing).
Yellow jersey Vincenzo Nibali (Astana) finished safely despite being held up by the incident, the Italian retaining a 7:10 lead over Frenchman Thibaut Pinot (FDJ) at the top of the overall standings.
Just 15 seconds separates Pinot from his countryman Jean-Christophe Peraud (Ag2R-La Mondiale) and fourth-place Spaniard Alejandro Valverde (Movistar) ahead of Saturday's decisive 54km individual time trial to Perigueux.
FIVE-MAN BREAK: An attack from the outset by Cyril Gautier (Europcar) sparked a response from Martin Elmiger (IAM Cycling), Arnaud Gerard (Bretagne-Seche) and Slagter, with Rein Taaramae (Cofidis) joining soon after to create the day's five-man break.
The leaders never saw their advantage rise about the four-minute mark as the rain lashed the remaining 164 riders in the famous Dordogne wine region of south-west France. With the gap reduced to just over a minute by the efforts of Sagan's Cannondale team, Slagter decided to break clear of his fellow escapees with 22km remaining.
The Dutchman held just a 20-second gap as the decisive climb approached, thinning out the peloton which was roaring along at a ferocious pace. Garmin placed a block of riders on the front as Navardauskas was slingshot up the hill - catching Slagter just over the crest before powering clear on his own individual time trial towards the finish.
"It was planned to be in the break - and to win the stage - but we often make such plans and we certainly didn't think it would work out like it did," said Slagter, who won the prize of the most aggressive ride of the day. "On the climb I saw Ramunas come and I thought he had a chance. It was incredible."
Once their man was ahead, Slagter and his Garmin team-mates rode en bloc on the front of the chasing pack, disrupting the pace and reeling in any counter-attacks. It was an expert all-round performance from a team whose main rider - Andrew Talansky - was forced to withdraw in the second week following a spate of crashes.
Talking of crashes - it's hard to tell whether or not Navardauskas would have gone the distance had the chasing pack not been disrupted by a huge pile-up coming off the bridge over the river Garonne into the centre of Bergerac. Sagan and the German national champion Andre Greipel (Lotto Belisol) were caught up, while the other sprinters were forced to fight for scraps.
Degenkolb pipped double stage winner Kristoff for second, with Mark Renshaw (Omega Pharma-Quick Step) and Daniele Bennati (Tinkoff-Saxo) completing the top five.
RIDE OF THE DAY: Besides Garmin's tour de force - from Slagter's role in the break, right through to the pace-setting up the Cote de Monbazillac and Navardauskas's finishing touches - special mention must go to Swiss national champion Elmiger, who entered the stage having already clocked up 611km in breakaways since the start of the Tour in Yorkshire. He added another 190-odd to break the 800km barrier - which warrants a 'chapeau' in anyone's books.
DAY TO FORGET: John Degenkolb finally won a bunch sprint - but it was only for second place. At least the German could even do that: Peter Sagan will now have to win on the Champs Elysees if he wants to break his duck and leave the 2014 Tour as a stage winner.
COMING UP: T he time gaps could well be huge for the rolling 54km race against the clock from Bergerac to Periguez on Saturday. If Tony Martin is the outright favourite, there is one other near-certainty: the 15 seconds that split Pinot, Peraud and Valverde going into the time trial will stretch out considerably over the course of the afternoon. The question is - which two of these three riders will flank Vincenzo Nibali on the podium in Paris?
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