Radwanska headed into the contest at the Singapore Sports Hub with an inferior 5-1 record against the Wimbledon champion, but she was in inspired form to win her White Group opener in just 90 minutes in the round-robin format.
By contrast, world number three Kvitova, WTA Finals winner in 2011, put in a disjointed performance as 26 winners were soured by a costly 42 errors and she now faces an uphill task to qualify from the group.
"It's never easy against her [Kvitova]," Radwanska said after her win.
"She's a great champion and has been playing great tennis in recent months.
"I'm just happy I can win. There are just eight of us here and you do your best to try and progress."
Radwanska had the first break point of the match when Kvitova put a backhand into the net and she clinically dispatched the chance with a fierce volley at the net to move into a 4-2 lead.
The Pole fought back from 0-30 down in the next game to go 5-2 up and saw a set point go begging at 30-40 on Kvitova's serve.
However, Radwanska was afforded another chance when Kvitova went long with a forehand and this time she made no mistake with a thumping forehand that her opponent was unable to return.
Kvitova improved significantly at the start of the second set, though, and a delightful passing shot brought a second break point at 1-1, although Radwanska produced a sublime backhand drop-shot and recovered to hold.
The world number six had no such luck on her next service game, however, a thumping forehand from Kvitova earning a break point that was duly taken for a 3-2 lead.
That advantage was immediately wiped out. Kvitova fended off three break points to restore deuce, but an audacious lob earned Radwanska a fourth opportunity that she duly seized.
The momentum continued to swing back in Radwanska's favour as she fought off a break-point chance for Kvitova to lead 4-3.
Radwanska followed up in the next game with a controlled overhead volley that brought two chances to break and she needed just one to have the chance to serve for the match.
Kvitova fought hard to prolong the encounter, but ultimately Radwanska secured the win with her third match point.