Quartararo finished third in a race that was suspended for half an hour due to an early crash, which saw two bikes engulfed in flames.
Both riders – Dani Pedrosa and Lorenzo Savadori – escaped relatively unscathed, though the latter was unable to restart the race.
The restart benefited Martin, who had set a record time at the Red Bull Ring to take pole position in Saturday's qualifying session.
Behind Martin, Joan Mir (Suzuki Ecstar) claimed second, with Quartararo nipping into third – Miller's hopes fading when he skidded off the track at Turn 7 with 10 laps remaining.
Quartararo (172) now has a 40-point lead over Johann Zarco, who finished sixth, in the championship standings, with Mir further behind on 121.
"It feels so good because when there is a red flag it is always tough," said Quartararo.
"I was extremely good on the braking on Turn 3. I knew that there was a possibility to overtake and did it.
"I don't really know the position of the other guys apart from Joan and Jorge finishing in front. I am so happy. The best goal was to finish on the podium and we did it. So happy for Jorge, he took some points off Joan, so a good day."
Martin, meanwhile, was ecstatic after achieving what he labelled the first step on his path to pushing for world championship glory.
"I can't believe it, I don't believe it, that's why I'm not so excited! I think the ride was amazing. I kept a really constant pace. I was super focused," said the 23-year-old.
"Even if I did some mistakes, my target was to win the race. Joan was impressive today, he was behind me almost all the race but then in the last laps I tried a bit more and I took a gap for the lead.
"The last laps I was thinking about things – everyone who has helped me to be here – that's why I did worse in the last laps but I had this gap to manage. Thanks to all my family, this is one big step towards my big area to be world champion."
Reigning world champion Mir said: "Today was close. I'm happy because all weekend we have been there, overall.
"I needed a couple of faster laps to fight with Jorge, he did a great job. We must push a bit more, find something more for next weekend if we want to be on the top of the podium.
"I am proud of the team, they have done a great job and finally, the performance is higher, so happy and proud of them."