Kuznetsova, once ranked as high as No. 2, is a two-time Grand Slam winner. She lost to a player whose ranking climbed 120 places to 63 in 2012. Kuznetsova took an extended break after dropping the second set to receive treatment for a foot injury.
The wind on centre court at the Auckland Tennis Center contributed to more than 100 unforced errors. But Bertens kept her composure in recording one of her best victories.
Bouchard, the junior Wimbledon champion from Westmount, Que., dropped a 7-5, 6-3 decision to Russia's Elena Vesnina.
Unseeded American Jamie Hampton ousted defending champion Zheng Jie of China 7-5, 6-1. McHale, whose ranking peaked at No. 24 in August before slipping to 33 at year's end, also struggled in the wind.
In another upset, Pauline Parmentier of France ousted seventh-seeded Christina McHale of the U.S. 6-4, 6-1.
Former champion Yanina Wickmayer, seeded third, defeated Magdalena Rybarikova 7-6 (3), 6-2.
Greta Arn, the 2011 champion, lost 6-4, 3-6, 6-2 to Thailand's Nudnida Luanganam in the final round of qualifying but gained a place in the main draw as a lucky loser. Arn will face top-seeded Agnieszka Radwanska after the withdrawal of injured Frenchwoman Kristini Mladenovic.