The third seed made a shaky start to the match when she lost the opening game on her own serve, before Svitolina produced a love game to race into a 2-0 lead.
The Ukrainian looked to be on course for a shock victory when she broke again in the fifth game to move 4-1 ahead in the opener.
However, Radwanska fought back and won the next four games in a row to level things up again, with both players then holding serve to force a tie-break that the Pole eventually won at the third time of asking.
Svitolina continued to battle in the second set and almost made another early breakthrough when she ran Radwanska close in the second game, but the third seed overcame four break points to hold.
She then broke the unseeded Ukrainian in the very next game to take control of the match, but was pegged back immediately as Svitolina got a break of her own.
Svitolina then forced a deciding set with another break in the final game of the second, edging to a 7-5 win having denied Radwanska three game points.
Having been pushed to the limit during the first two sets, Radwanska's class and experience finally showed in the decider as two unanswered breaks helped the 2012 champion earn victory in 156 minutes.