Kvitova went down by a service break in the third game of the match, but avoided the upset bug that has sent every other seeded player in the tournament packing.
Trailing 4-3, the second seed called her coach onto the court, and he told her to stop trying to win on every shot and be more patient.
''So I was trying to play a little bit longer rallies, just wait for the good point that I can really attack her, going forward for the final volleys, for example,'' she said.
It worked. She broke back to tie it at 4, and then broke Zahlavova Strycova again to take the set.
She cruised in the second, taking the final five games of the match.
Kvitova, who won here in 2012 and lost in the finals last year to Simona Halep, said she's not treating this tournament as a glorified practice for next week's U.S. Open Slam- something other big names have been known to do.
''I want to play my best tennis and have the great result I can,'' she said. ''Every match I step on the court, I want to win.''
After struggling with Flipken's serve all match, Stosur broke her to go up 4-3 in the final set on a nice inside-out forehand, and then broke her again in the final game of the match.
''The last couple of games, I just tried to return a little bit smarter with where I was trying to place it,'' she said. ''Once I was able to hit a return to a better part of the court, then I felt like I was in control of the point more.''
On the other side of the bracket, Magdalena Rybarikova of Slovakia ousted American Alison Riske in three sets, 7-5, 0-6, 6-4. Rybarikova, who beat the top-seeded Halep in the second round, broke Riske's serve six times. But she failed to hold her own serve seven times, and said she felt lucky to escape with the win.
''I was really fighting, trying to somehow make her play worse, to make her to make more mistakes than me,'' she said. ''It kind of works.''
Riske, who was the last remaining American in the draw, said she made too many errors in the third set. Three double faults in one game put her down 4-2 and she was unable to climb all the way back.
''Obviously at this level, that can't happen,'' she said. ''That was a huge problem.''
Rybarikova will face hard-serving Italian Camila Giorgi in the semis. Giorgi, who upset Caroline Wozniacki in the second round, got by Garbine Muguruza of Spain, 6-4, 6-7 (4), 6-2.
Giorgi, who has climbed to No. 38 in the world rankings, double-faulted 19 times, but compensated with a strong return game, breaking Muguruza nine times.
''My type of game, it's more secure when my serve goes in, and it's easier,'' she said. ''But It's not a problem. ... I mean, I have the return.''