Mattek-Sands, a former top-30 player who has slipped to No. 197, is playing as a wild card. She eliminated the sixth-seeded Vekic in nearly two hours.
The American has had more success in doubles of late, winning her second title of the year with Sania Mirza in Dubai last weekend.
Also advancing were fifth-seeded Misaki Doi of Japan, Patricia Mayr-Achleitner of Austria and Karolina Pliskova of the Czech Republic.
Playing Wednesday are top-seeded Caroline Wozniacki and defending champion Hsieh Su-wei of Taiwan.