The 17-year-old, playing in only her second final, fought back from a set down to win 4-6, 6-4, 6-1.
The sixth seed was also a break down midway through the second set, but lost only two more games in the match to race to the finish line.
The 76th-ranked Anisimova will continue her rapid rise up the world rankings, having also impressed with a run to the Australian Open last 16 earlier this year.
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