The world number nine lost serve in the fourth game of the first set to allow her Serbian opponent to race ahead and eventually take the set.
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The second set went by the way of serve until the fourth game when Jovanovski lost out, allowing Azarenka to get a foothold on the contest and level the match overall.
Azarenka lost serve three times in the deciding set, but Jovanovski failed to capitalise by losing serve on two occasions herself.
The 22-year-old, who is currently ranked 45th in the world, did eventually prevail though, seeing out the set 7-5 to dump Azarenka out.