The Garmin-Sharp rider beat Astana's Jakob Fuglsang to the line in Bagneres-de-Bigorre after the pair had attacked off the front, five kilometres from the top of the final climb of the day, and then stayed ahead down the long descent into town on Sunday.
Birmingham-born Martin became only the fifth Irish cyclist to win a Tour stage, and the first since his uncle's victory on stage 16 of the 1992 Tour in La Bourboule.
"Every win is important and special in its own way," the 26-year-old said.
"It was such an incredible day today because this team Garmin-Sharp shows such a team spirit.
"Everyone gave 100% today and some of the guys nearly missed the time limit because they gave so much for my victory."
We decided this morning on the bus that I was going to try and win the stage and we've succeeded so it's incredible.
Martin and Fuglsang made their move as the main contenders at the front of the race, with Chris Froome successfully defending his yellow jersey against the likes of Alejandro Valverde and Alberto Contador.
Fuglsang led the pair into town and towards the finish line, Martin was close behind and, after some nervous glances between the pair, he kicked out 250 metres from the line and Fuglsang could not answer.
"It's hard to describe how it feels, it's more relief actually because I knew I was the favourite coming into the sprint and I was quite confident, but I still had to do it," Martin added.
"To come across the line knowing that I've won a stage of the Tour de France is amazing. In the end, the scale of the event wasn't on my mind - it was just another bike race.
"I was so focused on his wheel and beating that guy in the sprint that I didn't even look behind once to see where the peloton was. It was just a case of focus on the finish line and get there first."
Stormont Sports Minister Carál Ní Chuilín said: "Dan Martin has claimed a first stage victory on the Tour de France, a terrific personal achievement and Ireland's first Tour de France stage winner since 1992."