If there was ever any doubt that Laura Robson both has what it takes to go a very long way and is a big match player then her victory yesterday over yet another grand slam champion should set people straight.

It wasn't pretty and there were more errors than winners but the 18-year-old found her best form when she needed it to defeat Petra Kvitova, the 2011 Wimbledon champion, 2-6, 6-3, 11-9 in a match played in searing heat, which lasted three hours and finished well after midnight.

The win was worth a place in the third round of the Australian Open and, with Heather Watson due to play her third-round match in the early hours of this morning, it is the first time two British women have been to this stage in 22 years.

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Robson reached the last 16 at the US Open with wins over Kim Clijsters and Li Na but, if anything, her victory over Kvitova, especially without playing her best, was more satisfying.

"I feel I was playing better in New York," Robson said. "I thought today was pretty ugly but in terms of how tough it was to close it out in the end, I think it's right up there with one of my best wins."

Even at the age of 18 – she is 19 next week – when Robson is in full flow there are few better sights in women's tennis. Her left-handed serve is becoming more of a weapon and her forehand is a massive threat. But it is her movement that has also been a revelation in recent months. Fully fit after a series of niggling injuries, she has been working with new coach Zeljko Krajan, with the help of Andy Murray's fitness trainer, Jez Green.

A month spent in Florida helped get her used to warm temperatures, albeit not quite as extreme as yesterday when the mercury reached 105°F.

It was still over 90°F when Robson and Kvitova started and at times it was like looking in a mirror, so closely matched are they in terms of style. After two error-strewn sets, Kvitova led 3-0 and 4-2 in the third only for Robson to hit back and break for 6-5. She played a bad game to drop serve but from then on, she was rock solid and Kvitova's resistance finally broke in the 19th game of the deciding set, with Robson then holding to record another famous win.

Kvitova often struggles in hot conditions because she suffers from asthma but the Czech said it was only her serve – she had 18 double faults – which let her down.

"I don't think I've had a worse serving day than today," the No.8 seed said. "That was the key. I didn't play my best today, for sure. I was 3-0 up in the third set so I should have been able to hold that lead but I didn't."

Tomorrow, the crowds at Melbourne Park may get their first look at a clash which could go on to become a major rivalry when Robson takes on American Sloane Stephens.Serena Williams has tipped 19-year-old Stephens as her heir apparent while Robson has been singled out for big things since she won junior Wimbledon in 2008.

"I wouldn't say it's a rivalry but we're the same age," Stephens said. "It's not like Federer and Nadal – but it could be."

Robson and Stephens first met when the Briton was about 10 and their most recent meeting came in Hobart last week, where Stephens came out on top.

"She's in good form," Robson said. "She's a good mover, good ball-striker. It's always going to be a tough match but, hopefully, I can play better than I did today."

Defending champion Victoria Azarenka and favourite Serena Williams both cruised through to round three while the youngest player in the draw, 16-year-old Donna Vekic, was beaten 6-1, 6-4 by Caroline Wozniacki, the former world No.1.

However, the oldest player in the women's event lives on. The 42-year-old Kimiko Date Krumm, of Japan, who reached the semi-finals here in 1994, beat Israel's Shahar Peer 6-2, 7-5.