Ukrainian teenager Elina Svitolina toppled third-seeded Czech Kvitova 6-2, 7-6 (7/2). Bouchard went down to Russian veteran Svetlana Kuznetsova 6-4, 3-6, 6-2.
Bouchard has now dropped both of her summer hardcourt matches, also losing stumbling out of the gate at home in Montreal last week.
Kvitova's exit in her first match after enjoying a first-round bye came a week after a third-round exit in Montreal, and the world number four admitted she couldn't find her way against the Ukrainian, who is playing her first tournament since defending her title at Baku last month.
Kvitova held a 3-0 lead in the second set as the Czech attempted a comeback effort. But Svitolina, ranked 39th in the world, nudged the scoreline back to a 5-4 lead only to be broken as she served for the victory.
The pair traded two more breaks before the tiebreaker, where Svitolina claimed victory on the first of four match points as Kvitova mis-hit a return wide.
"From the beginning I didn't feel very comfortable out there," Kvitova said. "And I was trying to really fight, I just wasn't able to do that today."
Fourth seed Agnieszka Radwanska beat Japan's Kurumi Nara 6-2, 6-2.
Stosur, who beat Williams in the 2011 US Open final, made the world number one fight for her win.
"She was up in both of the breakers, it was just a great match, to be honest," said the winner. "She served unbelievable, it was really intense."
German Angelique Kerber, the sixth seed, advanced against Ekaterina Makarova 6-4, 6-1 while Italian 13th seed Flavia Pennetta beat Taylor Townsend of the US 6-4, 6-3 to set up a clash with Williams.