With the temperature rising to 109 degrees, the Extreme Heat Policy came into play at Melbourne Park and organizers closed the roof over Rod Laver Arena.
Williams, a three-time Australian champion, had dropped the first set before the roof was closed with the temperature nudging 107.
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She was only a game away from a quarterfinal exit in the second set before recovering to beat the 2004 U.S. Open champion 5-7, 7-5, 6-1.
When the air conditioning finally kicked in, so did Williams' performance.
Until then, she said, "I was in like an out-of-body experience."
"I felt I was watching someone play in a blue dress, and it wasn't me, because it was so hot out there," Williams said. "And I kept trying to tell myself that it's not hot, you know.
"But it got hotter ...."
She moved into a semifinal against Olympic gold medalist Elena Dementieva, who played 95 minutes in the open for a 6-2, 6-2 win over Carla Suarez Navarro of Spain.
The men's match was played entirely under the cover of a roof on Rod Laver Arena and Verdasco cashed in, converting all four of his break-point chances and fending off all but two of Tsonga's 13.
Verdasco, who had an upset win over Britain's Andy Murray in the fourth round, will play the winner of Wednesday's night match between another Spanish left-hander, No. 1-ranked Rafael Nadal, and No. 6 Gilles Simon of France.
Williams, who has nine Grand Slam singles titles, was down a set and a break and with Kuznetsova serving for the match when she started her comeback.
She broke even at 5-5 in the second when Kuznetsova missed an open-court volley that turned the match.
Williams held and broke Kuznetsova's serve again, pulling to a set apiece with a forehand winner down the line.
The American broke to lead 3-1 in the third and saved two breakpoints with a pair of forehand winners to take command.
After beating the Russian, she joins three others in the semifinals. No. 3 Dinara Safina plays No. 7 Vera Zvonareva on the other side of the draw.
"Me against the Russians, I guess," Williams said.
Dementieva and Williams will play the first of the semifinals Thursday, Safina and Zvonareva having had an extra day off after advancing on Tuesday.
Dementieva questioned the wisdom of leaving the stadium roof open for her match as temperatures soared. The heat eventually suspended play on outside courts.
"I finally started playing a little better in the end of the second set," Williams said of her comeback after the break, when she got her rackets restrung. "I felt like I was really, really off before."
Kuznetsova said she was angry the roof was closed.
"It gave her more chances," Kuznetsova said. "Definitely it was a big change. I was very comfortable playing outside.
"She has big serve. She was using it very good when the roof was closed. I guess it was in her favor very much.
"Definitely angry. Why should I not be?"
Williams won the Australian title in 2003, 2005 and 2007 and is now two wins from continuing the odd-numbered sequence.
"I just have to keep playing well and just go for two more," she said.
Not wanting to waste time in the sun, Dementieva raced to a 4-0 lead after winning 16 of the first 18 points against the 20-year-old Suarez Navarro, who'd upset Venus Williams in the second round.
After eventually holding serve in a sixth game that went to deuce 11 times and lasted 17 minutes, Dementieva finished off in 1:35.
"You can work so hard trying to get ready for the weather conditions, but when you have to face 40 (104 Fahrenheit) or 41 (106), there is no way you can get used to it," Dementieva said. "The best way is to play as quick as possible and just get away from the court. I mean, there is no way to adjust with the heat here."
Fourth-seeded Dementieva had never advanced past the fourth round in 10 previous trips to Australia. Her best Grand Slam performances came at the French and U.S. Opens in 2004, when she lost in each final.