Razzano, who needed a wild card to gain entry to the main draw, is ranked No. 131 in the world. Kuznetsova, who won the 2004 U.S. Open and the 2009 French Open, is 29th.
Razzano used her superior serve to control her Russian opponent, who had six double faults.
The fifth-seeded Stosur needed 1 hour, 28 minutes to beat the American after struggling through the opening set.
The 13th-ranked Stosur was granted a wild card after losing to Olga Govortsova last week in the opening round of the Bank of the West Classic. The Australian wanted to get more time on the hard courts in preparation for the U.S. Open, an event she won in 2011.
Stosur took advantage of Lepchenko's first serve by winning 53 percent of those points and breaking her serve five times.