The 18-year-old Robson, who beat Kim Clijsters and Li Na at last year's U.S. Open, took advantage of 18 double faults from former Wimbledon champion Kvitova to reach the third round.
"I knew it was going to be very, very tough and trying to return her serve in the first set was very, very hard," 53rd-ranked Robson told the crowd.
"But I knew once I got my returns firing and a bit more confidence on my serve I could get myself back into the match and that's what I did."
With temperatures still close to 90 degrees as the match moved past midnight, both players made a stack of errors but Robson recovered from 4-2 down to serve for the match at 6-5.
Nerves got to her and Kvitova broke back but Robson broke again in the 19th game and then held on to set up a clash with American 19-year-old Sloane Stephens.
Robson's win, combined with the progress of Heather Watson, means there are two British women in the third round in Melbourne for the first time since 1991. (Reporting by Simon Cambers; Editing by Martyn Herman)
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