British number three Jones broke into the top 100 last year, having won two WTA 125 titles – the rung below the main tour.
She was previously told by doctors she would not play tennis professionally, having been born with a rare genetic condition.
Jones has Ectrodactyly Ectodermal Dysplasia (EEC), which means she has three fingers and a thumb on each hand, three toes on her right foot and four toes on her left.
Jones struggled with the hot conditions in Auckland but battled back well, saving nine of the 13 break points she faced to advance in just over two hours.
“She was unplayable the first set and a half and I found it really tough to get used to playing in the day compared to the night. It’s a lot quicker,” Jones added.
Jones will play Chinese seventh seed Wang Xinyu for a place in the last four.