Garbine Muguruza trailed 4-0 in the second set, but won the next five games as Mona Barthel was unable to force a decider.
Jelena Ostapenko put her Australian Open woes behind her to defeat Kristina Mladenovic in a frenetic contest.
All eyes will be on Serena Williams and Simona Halep in round four of the Australian Open after each earned convincing wins on Saturday.
Garbine Muguruza and Johanna Konta served up an early-morning treat for those spectators who stuck around in Melbourne.
Two-time grand slam champion Garbine Muguruza has been forced to withdraw from the Sydney International because of illness.
The Hopman Cup final will once again be contested by Switzerland and Germany after the latter got the better of Australia in Perth.
Host nation Australia and Germany remain unbeaten at the Hopman Cup and will meet on Friday with a place in the final at stake.
Switzerland and Germany recorded Hopman Cup wins on Sunday, with Roger Federer, Alexander Zverev and Angelique Kerber among those to star.
Julia Goerges' WTA Elite Trophy defence was ended by Ashleigh Barty, who will face Wang Qiang after the Chinese crushed Garbine Muguruza.
Two-time major champion Garbine Muguruza was on the brink of being knocked out in Zhuhai, but rallied to reach the last four.