The Spanish 10th seed was struggling with an injury in the opening set, and was bagelled by Gavrilova in an extraordinary opening spell.
But Suarez picked it up, her knee clearly loosening up from the second set onwards as she won 0-6 6-3 6-2.
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Russian-born Gavrilova, whose Australian passport only came through last month, became increasingly agitated and frustrated as the game got away from her, despite warm support from the local crowd.
Carla Suárez v Daria Gavrilova: As it happened
But Suarez held firm and used her experience to keep calm throughout, earning a quarter-final clash with Agnieska Radwanska.
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