The windy and flat 217-kilometre (134.8-mile) run with one major climb from Saint-Paul-Trois-Chateaux to Le Cap d'Agde was tailored for a win by one of the race's sprinters.
The German's ability to get over the super-steep Mont Saint-Clair climb, in the picturesque port town of Sete, about 23 kilometres from the finish helped pave way for the victory. Several other top sprinters - like Britain's Mark Cavendish - struggled up the hill and fell back.
Greipel, who turns 30 on Monday, earned his third stage victory of this Tour after winning the fourth and fifth stages in sprint finishes. Still-photo imagery showed he won by half a wheel's length ahead of Slovakian rider Peter Sagan. Edvald Boasson Hagen of Norway was third.
Wiggins trailed close behind in the main pack. Overall, he leads his second-place Sky teammate - and fellow Briton - Christopher Froome by 2 minutes, 5 seconds. Vincenzo Nibali of Italy is third, 2:23 back, and defending champion Cadel Evans of Australia is 3:19 back in fourth.
Saturday's route was known as a transitional stage because it was mostly flat, and guided riders away from their last big test - the Alps - and toward their next, the Pyrenees.
Greipel's Lotto Belisol team did the hard work of leading the pack through a wind-swept ride along the shore in pursuit of breakaway riders Michael Albasini and Alexandre Vinokourov, ultimately catching them.
In a bold move, with less than a kilometre left, Wiggins powered up to the front of the pack with Sky teammate Boasson Hagen on his back wheel, trying to set up the Norwegian for the stage win.
Greipel said he "speculated" that such a plot was being hatched. He pulled up just behind Hagen, then whizzed around him after a final bend and held on to the line.
"I'm really happy with this victory... it was once again a team effort," said Greipel, who has four career Tour stage wins. "The sprint was very long. I was just on the wheel of Boasson Hagen, and I saw that I could win if I just gave a little extra at the end."
Wiggins said he led the late surge because he wanted to stay in front and out of possible trouble in a big final bend in the road. He also wanted to help Boasson Hagen to return a favour for his support in the Alps.
"Once we knew that Cavendish wasn't going to come back, everybody said we'd try to do the job for Edvald," Wiggins said. "Sometimes it's just good to get on the front and try to repay a friend of mine back."
Still, Spain's Luis Leon Sanchez, who had been leading in a two-man breakaway that was overtaken by Wiggins, showed his frustration with an angry hand gesture against the man in the yellow jersey.
Sanchez said he felt Sky was being too dominant and didn't understand why Wiggins was working to help a teammate win a stage while in yellow himself.
"It's unfortunate. I can't look after everyone in the peloton," said Wiggins, adding that he does "love" Sanchez as a person. "It's a shame he feels like that."
Sanchez later appeared to have second thoughts about his first reaction and wrote on Twitter, "I beg (at) bradwiggins' pardon. He is the leader of the race so he and his team have the right to do whatever they want."
As the riders neared the coast, the stage's big challenges arrived, including a windy ride between the Mediterranean Sea and the Bassin de Thau - known for its oyster farms.
Riders swung their bikes left to right as if in slow motion as they arrived at Mont Saint-Clair, a mid-grade ascent over 1.6 kilometres - but with an average gradient of 10.2 per cent.
Evans attacked on the Mont Saint-Clair and got a few seconds ahead of Wiggins. But the Briton and his other closest rivals for the yellow jersey all hung close on the Australian's back wheel.
However, the climb and the windy road to the finish split up the pack, with sprinters like Cavendish and Australia's Matt Goss in a bunch that was more than 8-1/2 minutes behind Greipel and the overall race leaders.
Sunday's 14th stage takes riders along two big climbs in a 191-kilometre trek from Limoux to Foix.Suggest a correction