The 22-year-old Czech player, who was seeded eighth, was beaten 2-6 6-3 11-9 by the British number two on Rod Laver Arena in the latest-finishing match of the tournament thus far.
Robson started the match with back-to-back double faults and was broken to love in the opening game.
However, the Melbourne-born Brit earned an immediate reprieve as she broke to deuce to level the set in the following game.
An unflattering first set, which was full of unforced errors from both players and saw a total of 11 double faults, eventually went the way of Kvitova when she converted set point number five on her own serve.
Kvitova continued to produce unforced errors throughout the second set. However Robson managed to cut down, allowing her to level the match at one set all.
The deciding set started badly for Robson, whose failure to find first serves saw her broken to 30 in game two.
Aided by three double faults from Kvitova in game five, Robson broke back to 40, but she then surrendered her serve again in the following game to leave Kvitova 4-2 ahead.
A frustrated Robson responded in the best way possible as she broke to love before holding to 30, levelling the final set at four games all.
The next two games went narrowly with serve, but Robson then edged ahead by breaking to 30, putting herself in a position to serve for the match.
In searing heat and under severe pressure, Robson failed to close out, as she lost her service (to 15) for the seventh time in the match.
The next six games went with serve, before Robson broke to 30 again to give herself a second chance to serve out for a place in round three.
Robson held herself together this time around, holding to love to seal victory in exactly three hours, leaving both players on the verge of tears for different reasons.
Sloane Stephens will provide the opposition for Robson in the third round.