The 21-year-old came through a deciding tie-break to claim victory after nearly three hours.
The tie-break was as tight as the 12 games that preceded it but, after saving one set point with a big serve, Draper was unable to prevent Kokkinakis taking the second.
The 21-year-old was virtually untouchable on serve now while his heavy forehand was mopping up the ones that did come back.
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