The round-robin match was the first played at the championships, which divides the world's best players into two groups of four before a playoff.
The ninth-ranked Zvonareva capitalized on error-strewn play by Kuznetsova, who often screamed in frustration after making mistakes.
Later Tuesday, top-ranked Jelena Jankovic played Ana Ivanovic, and Venus Williams faced Dinara Safina.
Licht: Where Are The Goals?
Kuznetsova leads Russia on Fed Cup roster
I Wish I Wasn’t Black - Asprilla
