Golubic, 23, slugged it out against the 29-year-old Italian for three hours before securing a 6-7 (7/9), 6-3, 6-4 win.
She next faces Estonia's Anett Kontaveit.
Second-seed and defending champion Jelena Jankovic of Serbia cruised through after Sweden's Rebecca Peterson retired in the second set.
Former Wimbledon runner-up Sabine Lisicki of Germany beat China's Peng Shuai 6-4, 6-1.