The Serb fought back to overcome his emotions and move past Dolgopolov 2-6 6-1 6-4.
He was crying as he left the court to thunderous applause from a packed crowd.
Djokovic learned of his grandfather's death while he was practicing on Thursday morning.
He had said how close he was to his grandfather, to whom he dedicated his victory in the recent Sony Ericsson Open in Miami.
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