"A couple of us went off and I think Geraint [Thomas] went into the barriers," Froome said. "Luckily, I just went over the gravel, managed to turn around and get back in."
Froome leads the general classification by 12 seconds from fellow Briton Thomas, with Italian national champion Fabio Aru of Astana third, 14 seconds back.
Irishman Dan Martin, fourth overall at 25 seconds back, tried a late attack as the final kilometre approached, but the Quick-Step Floors man was quickly shut down by Sky.
Australian Richie Porte maintained fifth place, 39 seconds behind Froome.
Tour of California winner George Bennett was the first Kiwi home today in 19th, 50 seconds behind Calmejane, and improved another three spots to 22nd in the general classification.
The other three Kiwis lost ground on the field, perhaps saving themselves for tomorrow's grueling mountain stage. Rookie Patrick Bevin sits in 89th overall, after trailing home more than 20 minutes behind the leaders today.
Veteran Jack Bauer was another 10 minutes back, obviously nursing Quick-Step Floors team-mate and sprint leader Marcel Kittel home, with debutant Dion Smith finished in that same group.
Calmejane, of Direct Energie, kept his strategy close to his chest in the day's breakaway, but barely looked back, after pulling away in the 11.7km climb to La Combe de Laisia.
Gesink (LottoNL-Jumbo) chased hard, but the Frenchman had too much power in the finale.
Calmejane, almost came to a halt five kilometres from the line, however, due to cramp in his right thigh.
"It's huge. It was everything I was dreaming of," said Calmejane, who won a stage in last year's Tour of Spain. "When I had cramp in the finale, I decided to drop a gear so I could pedal softer.
"I suffered like never before."
Team Sky will be tested tonight (NZT), when the 181.5km ninth stage takes the peloton through the Jura mountains, with three lung-busting climbs in prospect before a dangerous descent to Chambery.
"Tomorrow will be a very very hard day, it will be very selective," Froome said. "It's going to blow the general classification right open."