Fifth seeded Kuznetsova, a U.S. Open and French Open champion, was beaten by Lucie Hradecka 3-6 7-6 (6) 6-4 in a marathon two hour, 25 minute match on centre court while 2010 French Open champion Schiavone was beaten 6-4 7-5 by Urszula Radwanska.
There were no such upsets for Wozniacki and Williams, however, as the Dane dispatched Israeli qualifier Julia Glushko 6-3 6-2 while third seed Williams was imperious in her 6-1 6-0 demolition of Slovakia's Jana Cepelova.
"Usually when you win easily everything is going right, Williams told reporters of her 52-minute match in which she broke Cepelova five times. "I would like to play as many matches as possible and thankfully I felt pretty comfortable on the court so that's a plus.
"I was happy with my serve. I felt comfortable at the net and I was pretty calm.
"Sometimes your mind can be going a thousand miles an hour, especially when it's the first match of the season, so for the most part I felt like I was playing in real time and not fast forward."
Wozniacki was not as convincing as Williams but the 68-minute match undoubtedly shook out any cobwebs and set up a second round clash with American teenager Taylor Townsend, who beat 2010 Auckland champion Yanina Wickmayer on Monday.
"It maybe wasn't the prettiest match, but I got through it and I won it," Wozniacki said afterwards.
"It's always tough the first match of the season, you just want to get going and into the groove. But I'm happy to be through and getting another chance to play out there."
(Reporting by Greg Stutchbury in Wellington)
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