Martinez put Inter in front in the 22nd minute with his fifth goal in his past four matches for Inter.
"I work hard for this," Martinez said. "I do on the pitch what the coach asks and I always try to improve my understanding with my teammates."
The Argentina international could have doubled his tally late on but his spot kick was parried by Roman Bürki.
The penalty was won by Sebastiano Esposito, who was brought down by Mats Hummels. The 17-year-old Esposito had troubled the Dortmund defense since coming off the bench as he appeared unfazed at making his Champions League debut, despite having only previously played 17 minutes for Inter's first team.
As Dortmund pushed for the equalizer it left space at the back, and Antonio Candreva capitalized on that to seal the result for Inter with a minute remaining.
Inter and Dortmund are level on four points in Group F, three behind leader Barcelona after the Catalan side won 2-1 at Slavia Prague.
"It extends our hopes that we can advance in the Champions League from a very, very difficult group like this," Inter coach Antonio Conte said. "I'm happy especially for my players because at the end of last year they struggled a lot to get a place in the Champions League and to live these nights, with a full stadium and the fans in full voice
"We wanted to get the win to honor what the team did last season and to give a message to ourselves, that we're on the right road, that we can have our say even in such a difficult group."
Dortmund coach Lucien Favre labeled the return fixture on Nov. 5 as a must-win match for his team.
"We know we're in a very difficult group," he said. "We played well. We conceded hardly anything to Inter and they scored their first goal on a small error from us. I'd say that up until the 70th minute or so Inter created almost nothing.
"There are only three matches left so everything is possible. We will play against Inter in our stadium and we definitely have to win that match."
Inter went into the match needing a win after defeat in Barcelona and a draw at home to Slavia Prague, while Dortmund had won in Prague and managed to hold Barcelona to a draw.
Inter broke the deadlock when Stefan de Vrij floated a great ball over the top for Martinez to spring the offside trap and fire into the back of the net.
There was a brief check with the video assistant referee before confirmation that Nico Schulz had played Martinez onside.
The majority of the stadium erupted into cheers, briefly quieting the Dortmund fans, who had been a noisy sea of yellow up in the third tier of San Siro.
It was the first goal the German side had conceded in this season's Champions League.
Dortmund almost levelled with the last kick of the first half but Inter goalkeeper Samir Handanovic pulled off a great save to deny Jadon Sancho.
Candreva could have doubled Inter's lead almost immediately after the break but he fired narrowly over from inside the penalty area.
He did get his goal with a minute remaining, and shortly after Martinez's missed penalty, when Inter went on a swift counter and Candreva was sent clear by Marcelo Brozovic.