The 86th-ranked Meusburger broke Knapp in the opening game but missed two chances to go 4-1 up. After four consecutive breaks, Meusburger held serve at 5-4.
The Austrian failed to serve out the match at 5-2 but broke Knapp immediately afterward when the Italian ended the match with two double faults.
"I didn't play spectacular tennis, I just tried to be consistent," said Meusburger, who hasn't dropped a set in the tournament. "I kept the ball in play and waited for her mistakes. These tactics have paid off."
In a match that included 20 breaks of serve, Hlavackova got treatment for a right ankle injury toward the end of the opening set. She came from 4-1 down to take the second into a tiebreaker before breaking Svitolina's serve three more times in the decider.