Russia's Kuznetsova, a two-times grand slam champion who was knocked out in the first round in Auckland last week and had to qualify for the main draw in Sydney, beat the former world number one 7-6 1-6 6-2 in a match that lasted almost four hours.
Sydney had been forecast to hit a maximum of 43C on Tuesday as Australia swelters in a heatwave that has sparked raging bush fires.
The Australian Bureau of Meteorology reported a temperature high of 41.4 C at 1530(local) at Sydney's Olympic Park and women's top seed Radwanska had said earlier that play should have been abandoned until the temperature dropped.
The world number four, who was given a bye into the second round, played the opening match on centre court and beat Japan's Kimiko Date-Krumm 6-4 6-3 to advance to the quarter-finals but even that early in the day the heat was effecting the players.
"I think this is too hot to play tennis," Radwanska said. "Even for players, for ball kids, for the people sitting out there, I think it's just too hot."
"Today was one of the hottest day I (have) played for sure."
Former French Open champion Li Na, who followed Radwanska on to court, compared the heat to playing in a sauna, but she was good enough to speed through Japan's Ayumi Morita 6-1 6-0 in 52 minutes.
"I was feeling, I don't know, just feeling like unbelievable. They say it was 40, but even on the tennis court even more," Li said.
"I mean, even didn't finish the match, just finish (the) first set (and) I was feeling my feet already burning."
Kuznetsova's victory in the heat gave the 27-year-old Russian tremendous confidence she could continue to build towards the Australian Open having spent six months out rehabilitating a knee injury.
The former US and French Open champion spent two months on crutches last year and had to rebuild muscle strength in her legs to ensure she did not re-injure her right knee and dropped to 85th in the world rankings.
She had asked for a wild card for Sydney but was forced to play qualifiers, which had actually helped her build momentum into Tuesday's win.
"It was very weird to go and play qualifiers, but, you know, it just makes you stronger," she said.
"(It) definitely was a little bit uncomfortable, but it made me stronger (and) this is what I really was looking for in Sydney, to get matches."
Kuznetsova, who will meet second-seeded German Angelique Kerber in the quarter-finals, said she was not taking the win over the 10th-ranked Wozniacki as a turning point in her rehabilitation.
"I just don't think about it when it's turning point and when it's not," she said. "I'm here. I'm doing the best I can. That's it. That's over here. Tomorrow is another day."
WTA Sydney results
Madison Keys (US) beat Zheng Jie (China) 6-0 6-4
Svetlana Kuznetsova (Russia) beat 7-Caroline Wozniacki (Denmark) 7-6(4) 1-6 6-2
Roberta Vinci (Italy) beat Jelena Jankovic (Serbia) 3-6 6-4 7-6(4)
2-Angelique Kerber (Germany) beat Galina Voskoboeva (Kazakhstan) 6-2 7-5
Dominika Cibulkova (Slovakia) beat Ekaterina Makarova (Russia) 7-6(3) 1-6 7-6(1)
4-Li Na (China) beat Ayumi Morita (Japan) 6-1 6-0
3-Sara Errani (Italy) beat Maria Kirilenko (Russia) 6-1 6-1
1-Agnieszka Radwanska (Poland) beat Kimiko Date-Krumm (Japan) 6-4 6-3
WTA Hobart results
8-Sloane Stephens (U.S.) beat Simona Halep (Romania) 6-4 6-0
9-Mona Barthel (Germany) beat Chanelle Scheepers (South Africa) 7-6(3) 7-5
Elena Vesnina (Russia) beat 4-Yaroslava Shvedova (Kazakhstan) 4-6 6-2 6-1
Lauren Davis (U.S.) beat 2-Sorana Cirstea (Romania) 6-1 6-3
Bojana Jovanovski (Serbia) beat Maria Teresa Torro (Spain) 6-0 7-6(7)
3-Klara Zakopalova (Czech Republic) beat Timea Babos (Hungary) 6-4 6-3