Dumoulin said his bike broke when he knocked into Bardet near the foot of the second of two two-kilometre ascents up the Mur de Bretagne to the finish line.
"Having already lost 53 seconds on the stage - finishing 46th after a big effort from team-mate Soren Kragh Andersen, in the white jersey, to minimise their losses, he now drops to 19th - 1'23" behind yellow jersey Greg van Avermaet.
"My first thoughts are that I hope my wife hasn't gone into labour", smiled Martin as he caught his breath.
Britain's Adam Yates finished sixth on the day in the pack three seconds back, and has vaulted above Froome in the overall standings.More news: President tears into May's Chequers deal
Dumoulin and his Sunweb team compounded the mechanical mishap by getting a 20-second penalty when Dumoulin stayed too close to the team auto while trying to catch the pack, thereby benefiting from its draft that pulled him along.
Dutchman Tom Dumoulin, the 2017 Giro d'Italia champion, suffered a puncture with five kilometres to go and ended the day 53 seconds behind the victor.
The pace of the peleton quickened ahead of the category three climb to Mur de Bretagne, with the gap to the front shortening to under a minute with 20km to go.
"I saw the moment".
"It was an early break but it was my only opportunity for the stage win as I'd never have got it in a sprint", explained Martin.More news: China Trade Talks Come to Halt
"I was a bit nervous because of the headwind, I didn't think it was going to happen", Martin, 31, said.
A breakaway group of five riders - Laurent Pichon (Fortuneo-Samsic), Anthony Turgis (Cofidis), Dion Smith (Wanty-Groupe Gobert) and Damien Gaudin and Fabien Grellier (both Direct Energie) - hit the front from the off and built a gap of nearly seven minutes before it stabalised.
Friday will see the riders take on the longest stage of the race at 218km when they take off from Fougeres and finish in Chartres.More news: Frozen vegetables recalled from Aussie supermarkets over contamination fears