What had once been viewed as an opportunity to make time had become a chance to gain it back after Froome lost 51 seconds to his main rivals on Saturday.
BMC clocked 38min 46sec as their Belgian rider Greg van Avermaet took the overall leader's yellow jersey, with Sky in second place, four seconds behind.
"It's a dream. We rode a good time trial and the whole team worked hard", said Van Avermaet. "Just going on the feeling, I think we can be pretty happy", Froome said.
Overnight leader Peter Sagan was dropped by his Bora Hansgrohe teammates and fell to 80th overall, three minutes behind.
Froome's misfortune is a boost for France's Romain Bardet, seen along with Quintana as the four-time champion's main rival in this year's race, and also for Dutchman Tom Dumoulin, the 2017 Giro d'Italia victor.
Chavanel led for most of the race and did enough with his solo breakaway to secure the three-second time bonus, however he was eventually swallowed up by the chasing pack.More news: Beyonce, Jay-Z, Ed Sheeran to Headline Global Citizen Festival
Other top contenders Briton Adam Yates and Australian Richie Porte also lost 51 seconds after being involved in pile-ups.
Thomas missed out on the yellow jersey by just three seconds, having started the day sixth in the general classification and one second up on Van Avermaet.
"A bit disappointed not to win but it was a technical race because of the wind, changes of direction, up and down as well".
The Slovakian takes the stage and the yellow jersey with the Tour de France producing another compelling finish after yesterday's dramatic first day.
"It's amusing because at the end of the day it's just another bike race".
Among Froome's main rivals, France's Romain Bardet lost 1:15 today and now trails the Briton by 20 seconds.More news: Second rescue attempt begins after four boys pulled from Thai cave
Froome was left 55 seconds behind in the overall standings with another week of nervy rolling stages before hitting the Alps. "We gave it everything we had and it all went pretty much to plan".
"There's a photo of the team on the podium with Greg in yellow and I think it's one I will cherish for a long time".
Two-time runner-up Nairo Quintana is 1:31 back after he lost time following the puncture of both his tires near the end of Stage 1.
Among the big losers today were Nairo Quintana, whose Movistar team came home 53 seconds down on BMC Racing, Bahrain-Merida with 2014 Tour de France victor Vincenzo Nibali, 1 minute and 6 seconds down, and Romain Bardet's AG2R-La Mondiale, who came home 1 minute 15 seconds behind the day's victor.
Tuesday's fourth stage takes the riders from La Baule to Sarzeau over a largely flat 195km course.More news: Deadly protests as Haiti fuel prices spike