Garcia, who beat four-time Grand Slam champion Naomi Osaka within the earlier spherical, will play American Danielle Collins or Romania’s Sorana Cirstea for a spot within the semi-finals.
Elsewhere, Kazakhstan’s world quantity 4 Elena Rybakina progressed with a 6-3 7-5 win over American Madison Keys to arrange a gathering 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 via to the fourth spherical on the event for the primary time by beating Brazil’s Beatriz Haddad Maia in straight units.
The Briton broke again to disclaim Azarenka as the previous world primary served for the primary set, earlier than saving two break factors to degree the opener at 5-5.
However three-time Miami champion Azarenka struck once more two video games later, breaking Boulter to like to clinch a decent first set.
The second set proved rather more one-sided as Azarenka stormed right into a 5-0 lead with a double break earlier than Boulter was in a position to get on the board within the sixth sport.
Boulter managed to avoid wasting two match factors however she couldn’t deny Azarenka, whose upcoming opponent Putintseva beat Ukraine’s Anhelina Kalinina 6-4 7-6 (7-5).