World number nine Ashleigh Barty was trailing Kuzmova at the end of the first set, but the Australian rallied back to win 4-6, 6-4, 6-3 to set up a showdown against either Belinda Bencic or Kristina Mladenovic.
Meanwhile, the withdrawal of Serena Williams was accompanied by two high-profile retirements from the women's singles draw at the Internazionali d'Italia on Wednesday (AEST), before defending champion Elina Svitolina was eliminated in stunning fashion by Victoria Azarenka.
Either side of Williams pulling out of her scheduled second-round match with sister Venus due to an ongoing left knee problem, Caroline Wozniacki and Jelena Ostapenko were each forced to call an early halt to their respective contests in Rome.
Svitolina - seeking a third title in a row in Rome - then blew a 4-1 lead in the third set of a rain-hit duel with Azarenka, as the latter saved a match point and completed an unlikely comeback to triumph 4-6 6-1 7-5.
Wozniacki retired from her clash with Danielle Collins after losing the first set in a tie-break, a calf injury the issue on this occasion after back trouble prevented her from completing a match against Alize Cornet in Madrid last week.
Ostapenko, meanwhile, was a game away from defeat against Mihaela Buzarnescu when she stopped the tie, the Latvian later citing a fever as the reason.
Aryna Sabalenka and Anastasija Sevastova were beaten by Cornet and Belinda Bencic respectively.