Garcia, who beat four-time Grand Slam champion Naomi Osaka in the previous round, will play American Danielle Collins or Romania's Sorana Cirstea for a place in the semi-finals.

Elsewhere, Kazakhstan's world number four Elena Rybakina progressed with a 6-3 7-5 win over American Madison Keys to set up a meeting with Greek eighth seed Maria Sakkari.

Sakkari moved in to the quarter-finals with a walkover after Russian opponent Anna Kalinskaya withdrew.

Boulter had made it through to the fourth round at the tournament for the first time by beating Brazil's Beatriz Haddad Maia in straight sets.

The Briton broke back to deny Azarenka as the former world number one served for the first set, before saving two break points to level the opener at 5-5.

But three-time Miami champion Azarenka struck again two games later, breaking Boulter to love to clinch a tight first set.

The second set proved much more one-sided as Azarenka stormed into a 5-0 lead with a double break before Boulter was able to get on the board in the sixth game.

Boulter managed to save two match points but she could not deny Azarenka, whose upcoming opponent Putintseva beat Ukraine's Anhelina Kalinina 6-4 7-6 (7-5).