Irish rider Martin broke with one kilometer remaining on the final climb with French rider Pierre Latour offering the only competition.
Having been boxed in the last time the Tour finished on the Mur three years ago, the punchy climber made no mistake this time when jumping off the front of the peloton with 1,200m to go and opening a gap on the highest point of the climb.
This means that after six stages Dumoulin is now 1 minute 23 seconds behind current race leader Greg Van Avermaet (BMC Racing).
"I came to this team to be the leader and the pressure at the beginning, of being the leader is why I didn't do so well in spring", said the UAE captain Martin.
Spanish veteran Alejandro Valverde (Movistar) led a select chasing pack over the line for third place in the 181km stage from Brest, three seconds back, with BMC's Van Avermaet finishing safely in the mix to retain his lead in the general classification.
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.
It was in 2011 when the Tour first finished a stage on the steep, two-kilometre climb to Mur-de-Bretagne.
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 race went so hard on the first part of the climb, I saw everyone was on the limit and there were no teammates left, so why not have a try?"
"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.
"So I set off, kept my head down, kept going and my legs felt good".
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.