With rain pouring for much of the 208.5km stage between Maubourguet Pays du Val d'Adour to Bergerac, the Garmin-Sharp rider made a break in the final 15km to come home seven seconds clear.
Giant-Shimano's John Degenkolb was second, while Alexander Kristoff (Katusha) was third.
The yellow jersey holder, Vincenzo Nibali of Astana, was 24th and is expected to comfortably hold his position as the overall race leader in the final two stages to come.
There was a heavy crash involving several riders towards the end of the stage, involving AG2R La Mondiale's Romain Bardet and overall points leader Peter Sagan of Cannondale, but as the incident happened within the final 3km there will be no time implications for anyone involved.
Bardet later confirmed to the Tour's official website he had sustained no damage as a result of the collision.
Nibali's overall lead remains at over seven minutes from FDJ.fr's Thibaut Pinot, who finished 18th, but the day belonged to Navardauskas.
The Lithuanian made a break on the climb towards Bergerac, and no rider was able to stay with him as he lofted his arms in victory upon crossing the line.
Earlier in the stage, the soaking conditions had caused a number of punctures in the peloton with Giant-Shimano's Marcel Kittel - one of the favourites to take the stage - among those affected.
The already-crowned king of the mountains Rafal Majka (Tinkoff-Saxo) was 116th, and the riders will now head to Saturday's penultimate stage, the 54km time trial between Bergerac and Perigueux.