Tearney rallied from a set down to beat Rob Reynolds 3-6, 6-2, 6-1 while Bettink also fought back from losing the first set to beat fourth seed James Meredith 3-6, 6-3, 6-2.
By contrast the women's quarter finals were fairly straight forward.
Top seed Joanna Carswell cruised past eighth seed Elys Ventura 6-1, 6-4 and will take on third seed Sacha Jones in one semi after the ex WTA tour player crushed fifth seed Oleksandra Kalachova 6-2, 6-0.
Fourth seed Katherine Westbury was equally impressive in thumping sixth seed Nina Paripovich 6-1, 6-1. Westbury will take on seventh seed Emilia Price for a place in the final after she beat Louise Waite 6-1, 6-4.
The tournament moves to the ASB Tennis Arena for Friday's semi-finals and Saturday's finals.
The first match on centre court between Jones and Carswell gets underway at 3pm followed by the two men's semi-finals. The second women's semi-final is on court 4 from 3-30pm.
Draws
Women's
Men's