Azarenka, who is coming off a third round loss at Wimbledon last month, committed seven double faults and had her service broken five times in the contest.
She showed life in the second set as she seized a 2-0 lead, but 23-year-old Croatian Vekic had her way in the end to set up a quarter-final match with Kristie Ahn.
Ahn reached the quarter-finals by converting seven break points to upset third-seeded Belgian Elise Mertens 6-3 6-3.
Fourth seed American Amanda Anisimova, a 17-year-old, beat compatriot Madison Brengle 6-2 6-2 to reach the next round matchup with China's Saisai Zheng who routed Danielle Collins 6-2 6-0.