The American second seed, 20, has so far risen to 12th in the world in her breakout season and battled past Germany's Petkovic in over two and a half hours.
"It was definitely tough for a second round match," Stephens told the WTA website. "I thought I played well. Whenever I play Andrea, we always have good matches, so it was good to get through that."
"I was excited this week when I moved from 13 to 12. Just trying to take it slow and keep working on my game and improving.
"Obviously there's a long way to go, but I'm excited to keep going."
Stephens will face Stefanie Voegele of Switzerland in the last eight after she beat Karin Knapp of Italy 4-6 6-2 7-5. Stephens beat Voegele in Tokyo last month.
Slovakian Dominika Cibulkova thrashed Katarzyna Piter 6-0 6-1 in 52 minutes to reach the quarter-finals in Austria.
Poland’s Piter, playing in her first WTA event of the year, broke the seventh seed in the second set for her only game.
Former Linz finalist Cibulkova will face twice champion Ana Ivanovic of Serbia for a place in the semis.
Local hope Patricia Mayr-Achleitner dropped two match points at 5-3 in the second set before defeating Maryna Zanevska of Ukraine 6-1 6-7 (1) 6-4.
She will face number one seed Angelique Kerber of Germany after she beat Romania’s Alexandra Cadantu 6-1 6-0.
If Kerber wins her next match, she will be confirmed at the WTA Championships in Istanbul. Should she lose, Roberta Vinci, Caroline Wozniacki and Stephens are all in with a chance.