The 12th-seeded Australian was forced to wait out an hour-long rain delay before her match and when it started she and 21st seed Kontaveit were only on the court for about five minutes before weather halted play again.
They returned about two-and-a-half hours later to play one more game and then headed back to the locker room for a third rain delay, this time for two hours.
Barty finally secured the win with an ace on her second match point, six-and-a-half hours after the original scheduled start.
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(Reporting by Frank Pingue in Toronto; Editing by Ian Ransom)
