Two-time Wimbledon champion Kvitova, seeded 12th, needed just 59 minutes to get past Poland's Magda Linette 6-1, 6-2.
The Czech, who won her first North American hardcourt title in Montreal in 2012, didn't face a break point against the young Pole, who pushed former world No 1 Venus Williams to three sets in Stanford last week.
Kvitova, playing her first tournament since Wimbledon, broke Linette twice in each set and finished her off with an ace.
Former US Open champion Stosur, seeded 13th, had to work a little harder to dispatch Britain's Heather Watson 7-5, 6-3.
After some first-set struggles, however, she broke Watson three times in the second to wrap up the win.
"To get through in straight sets, I'm pretty happy," the Australian said. "Overall I thought I served pretty well. There weren't a lot of points off my serve where she could dictate off the first shot."
A string of rain delays stopped play for a total of four hours.
