The British teenager defeated the Czech 2-6 6-3 11-9 in just over three hours.
The victory for the Melbourne-born teenager sees two Brits in the last 32 of the Australian Open for the first time since Jo Durie and Anne Hobbs reached round three in 1987.
"I could definitely hear the crowd, I’m pleased so many people stayed," she said after the match that finished well after midnight local time.
"I knew it was going to be really tough, I always struggle to play lefties and I really struggled to return her serve in the first set but in the end I got through."
The eighth seed broke Robson in the opening game of the match and despite an immediate break back from the 18-year-old, the 2011 SW19 champ broke again in the fifth and seventh games to take the first stanza in 38 minutes.
But Robson showed the fight that saw her defeat Kim Clijsters at the US Open last year by breaking Kvitova in the fourth and sixth games of the second set to roar into a 5-1 lead. Despite a slight wobble, the 53rd ranked Robson took the set 6-3 in 48 minutes.
Kvitova surged into a 3-0 lead in the deciding set, but the player who complained of dizziness during her first-round win over Francesca Schiavone was seeing stars again as Robson broke back in the fifth game. A quick exchange of breaks followed before Robson finally held to level the match at 4-4.
The Fed Cup winner saved a break point in the ninth game with an ace to leave her London-based opponent serving to stay in the tournament. She duly did and then the 22-year-old from Bílovec hit a forehand long to hand the former Wimbledon junior champion a 6-5 lead.
Robson failed to serve out the match, being broken to 15, and then failed to capitalise on another break point at 7-7.
But when her next chance came she took it with both hands, lashing a forehand winner down the line to break in the 19th game of the third set and then serving out the match to join compatriot Heather Watson in the third round.
Robson will face American Sloane Stephens in the last 32, while Watson faces fourth seed Agnieszka Radwanska on Friday.