The Vacansoleil-DCM rider jumped out of a whittled down lead group with around 500m to go on the 3km finish climb and crossed the line six seconds ahead of Alejandro Valverde (Movistar) and Wiggins.
A 10-second bonus for the Dutch stage winner takes Westra to second place on GC, six seconds behind the Team Sky captain.
"It's a big day in my career. I worked hard for it and today was my day," Westra told a news conference.
The six-climb, 178.5km fifth stage from Onet-le-Chateau saw an early breakaway with Simon Clarke (GreenEdge), Frederik Veuchelen (Vacansoleil), Yukiya Arashiro (Europcar) and David Le Lay (Saur-Sojasun) breaking free after just 2km of racing.
They built up a lead of 6:40 on the bunch but with inside 10km remaining, and shortly after being caught by Kevin Seeldrayers (Astana) and Laurens Ten Dam (Rabobank), they were absorbed by the peloton.
With 13 riders within one minute of Wiggins on GC overnight, Team Sky's Richie Porte stymied any early breakaway attempts on the Mende climb, with ramps as steep as 10.5 percent, before Arnold Jeannesson (FDJ-BigMat) made a bid for glory just after the Flamme Rouge.
His attempt was short lived however as 29-year-old Westra powered away for the biggest victory of his career with Valverde, Wiggins, Leipheimer (Omega Pharma-Quick Step) and Simon Spilak (Katusha) awarded the same time behind.
American Tejay van Garderen, one of the Briton's main rivals, was dropped and finished 30 seconds behind Westra.
"It was a day in hell because of the cold," Wiggins said. "It was a bit warmer in the final climb and I have to thank my team mates Richie Porte and Rigoberto Uran for their hard work.
"The priority is still the final time trial but maybe we did the hardest part today. It's not the kind of climb I like but I have worked very hard on that this winter. It is very promising."
Friday's sixth stage will take the peloton over 178.5km from Suze-la-Rousse to Sisteron.