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The Romanian fought back from the brink at 11-10 in the epic final set before serving out for the match on the fourth attempt at Rod Laver Arena.
Halep closed out the dramatic contest on the first match point, when Davis, who had gone for broke all match, fired into the tramlines.
Halep will meet the winner of Ashleigh Barty and Naomi Osaka for a place in the quarter-finals.