Norway's Alexander Kristoff was second and Frenchman Samuel Dumoulin took third place as Italian Vincenzo Nibali retained the overall leader's yellow jersey.
"It wasn't a quiet stage at all. On paper it should have been like that but nervousness was always there in the peloton because of the wind," said Nibali.
"In the finale in particular, we could feel it but I was well covered by my team."
Lotto-Belisol rider Greipel, who had a mediocre start to the Tour, struck some 200 metres from the line as the peloton looked disorganised after Kittel, winner of three stages this year, dropped out of contention in the final kilometre.
His Giant-Shimano team first said he suffered a mechanical problem, but sports director Christian Guiberteau said later that Kittel was tired.
"It was a bit difficult for Marcel today. It was not just the aftermath of his fall (on Wednesday), he just did not have the legs today," he explained.
"It's not a mechanical (problem) or a puncture. It was just not his day."
Greipel said: "I felt a lot of pressure after the first few stages but finally we have a victory.
"It's a good answer from Lotto-Belisol to the critics. My confidence was always there. We stayed calm and did really good work. My team mates and myself, we deserve this win."
Asked about his relationship with Kittel, who had the upper hand in the first week of the Tour, he said: "We are rivals but we have a lot of respect for each other."
Australian Richie Porte, the new Team Sky leader after defending champion Chris Froome crashed out on Wednesday, and Alberto Contador lost key team mates as Spaniards Xabier Zandio and Jesus Hernandez both abandoned after falling.
"It's never good to lose a rider, but that's racing. Jesus Hernandez tried to continue but he really hurt his head," said Tinkoff-Saxo team manager Bjarne Riis.
Hernandez's head hit the ground, which left the Spanish climber shaken up.
"When I talked to him, he had the symptoms of a concussion. But he's on his way to the hospital at this moment and they will examine him so we'll know for sure," said sports director Philippe Mauduit.
"It's a sad day for Alberto and for team to miss a rider but we have to stay focused."
HOLLANDE VISIT
As French president Francois Hollande joined Tour boss Christian Prudhomme in his car, crosswinds on the Chemin des Dames ridge - scene of three World War One battles - split the peloton.
French champion Arnaud Demare and green jersey holder Peter Sagan were trapped behind but eventually made it back into the peloton, which was split again some eight km from the finish.
France's Thibaut Pinot, one of the riders with credible general classification ambitions, was caught behind and lost about one minute.
"It's not the end of the world," said the FDJ.fr rider, who is now 3:24 behind Nibali.
Another French contender, Romain Bardet, fell off and hurt his knee, although he lost no time on Thursday.
The Tour de France commemorated the centenary of World War One with the peloton's suiveurs (followers) invited to wear a 'Bleuet de France' cornflower in memory of the soldiers who died during the 1914-18 war.
The white jersey for the best Under-25 rider was also emblazoned with a Bleuet de France.
The peloton also paid tribute to past Tour riders, including winners Francois Faber, Oscar Lapize and Lucien Petit Breton, who died during the war.
Friday's seventh stage will take the peloton from Epernay to Nancy with two short categorised climbs near the finish.
