Stosur, playing for the first time since winning her first career Grand Slam title in New York, trailed in the second set but rallied to break her Russian opponent and take the final two games.
The two traded breaks in the third set, but Kirilenko broke again to make it 5-4 before serving out the match.
"You never want to go out in the first round after the week I had at the U.S. Open," said Stosur, who had a bye into the second round. "It's hard to explain how you could do well at the U.S. Open and then not do well here. I came as prepared as I could."
Stosur beat Kirilenko in the fourth round of the U.S. Open.
"I just played her a couple of weeks ago, so I knew what to do," Kirilenko said. "Every time you win a Grand Slam, it's tough to win the next tournament."
In the first round, Peng Shuai beat qualifier Jill Craybas of the United States 6-2, 6-0, American Christina McHale defeated Tamira Paszek of Austria 7-6 (1), 6-3 and Coco Vandeweghe of the U.S. won her rain-delayed match that started Monday over Zheng Jie of China 7-6 (4), 4-6, 6-2.
Third-seeded Victoria Azarenka of Belarus also advanced, beating Irina-Camelia Begu of Romania 6-1, 6-3. Azarenka, who has won two WTA Tour titles this season, saved nine break points and won 72 percent of her first serves.
Agnieskza Radwanska defeated qualifier Angelique Kerber of Germany 3-6, 6-3, 6-3, Ana Ivanovic of Serbia beat Laura Robson of Britain 7-5, 6-4 and Julia Goerges of Germany topped Greta Arn of Hungary 6-2, 6-1.