Former top-10 Slovakian Daniela Hantuchova drubbed the second-seeded Italian Errani 6-3, 6-2, while the sixth-seeded American Stephens stifled Spaniard Silvia Soler-Espinosa 6-3, 6-1 on the hardcourts at ASB Tennis Centre.
In other action involving seeds, No. 4 Czech Barbora Zahlavova Strycova held off South African Chanelle Scheepers 6-1, 3-6, 6-2 and Croat Ana Konjuh upended No. 8 German Mona Barthel 6-2, 6-1.
Also on opening day, Japan's Kurumi Nara doused Dutchwoman Kiki Bertens 6-3, 6-2, Lauren Davis overcame fellow American Shelby Rogers 1-6, 6-4, 6-3 and American wild card Taylor Townsend got past Belgian 2010 Auckland champion and two-time runner-up Yanina Wickmayer 7-5, 3-6, 6-0. Davis will meet another compatriot in Stephens in the round of 16.
On Tuesday, top-seeded former world No. 1 and 2014 U.S. Open runner-up Caroline Wozniacki will take on Israeli qualifier Julia Glushko and third- seeded former top-ranked star Venus Williams will battle Slovak Jana Cepelova.
Ana Ivanovic, who is not defending her title this week, topped Williams in last year's final in Auckland, which serves as a tune-up for the Australian Open, which begins January 19 in Melbourne.