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Robson, who only returned to action in June, was tested early in the first set on serve, but she was able to save four break points.
The 21-year-old improved from then on as she grabbed a crucial break point before eventually successfully serving to take the lead on Court 11.
In the second set, Vesnina fought back by snatching an early break, and despite creating three break-point opportunities of her own, Robson relinquished her lead.
With a big test facing her in the third set, Robson calmed the nerves and stormed to a 4-0 advantage, but Vesnina executed a break point to keep the Brit on her toes.
It seemed as though Robson's lack of match fitness took its toll as she was brought back to 4-4, but the Brit resumed focus to gain a break and give herself a chance of winning the match.
However, Vesnina turned the match on its head once again to successfully serve for victory.