Vesnina made a perfect start to the match when she broke Radwanska in the very first game before holding her own serve to race into a 2-0 lead.
However, the world number three soon fought back and recovered the break to level things up once again, missing a chance to move ahead when Vesnina rescued two break points.
The Russian broke again to regain the lead in the opener, but Radwanska responded immediately this time, winning against the serve to make it 4-4.
Vesnina saved two set points at 5-4 before squandering three break points of her own in a mammoth 11th game that Radwanska eventually won.
The Pole then clinched the opening set with her third break of the match, wrapping up a 7-5 victory at the first time of asking.
Vesnina again struck first in the second set, breaking in the third game to take the lead, but Radwanska was able to respond immediately once more.
Radwanska went on to win against the serve again in the eighth game before serving out the set for a 6-2 victory.