Russia’s Kuznetsova, the 2004 U.S. Open champion, served for the match in the second set but Williams rallied for a 5-7, 7-5, 6-1 win at Rod Laver Arena on Wednesday.
The stadium roof was closed after the first set when temperatures soared above 104 degrees and the tournament’s Extreme Heat Policy was put into effect, suspending play on outside courts.
The 27-year-old Williams has nine Grand Slam singles titles and won the Australian Open in 2003, 2005 and 2007.
Williams will play Elena Dementieva next.
Earlier, Dementieva got off to a fast start as temperatures soared, advancing to the Australian Open semifinals with a 6-2, 6-2 win over Carla Suarez Navarro.
The Olympic gold medalist wasted no time as temperatures hit 104 degrees, winning 16 of the first 18 points against the 20-year-old Spaniard.
She raced through the first five games in 22 minutes and, after eventually holding serve in a sixth game that went to deuce 11 times and lasted 17 minutes, finished off in 1 hour, 35 minutes.
Dementieva was having problems with her high service toss as she looked straight up into the midday sun, double-faulting four times in that long game. She finished with 10 double faults.
Both players were soon draping towels packed with ice around their necks during changeovers and sought shade as long as possible behind the baselines between points.
Suarez Navarro, who had an upset second-round win over seven-time Grand Slam winner Venus Williams, tied her best run at a major — her quarterfinal debut at the last French Open.
Fourth-seeded Dementieva had never gone past the fourth round in 10 previous trips to Australia. Now she’s only one win from reaching a third Grand Slam final. The 27-year-old Russian lost the French and U.S. Open finals in 2004.
Already set is an all-Russian semifinal between Olympic silver medalist Dinara Safina and Vera Zvonareva, who ran off the last 11 games to rout 2007 Wimbledon finalist Marion Bartoli 6-3, 6-0.
No. 3 Safina overcome 11 double faults and 36 unforced errors to end the surprising run of Jelena Dokic 6-4, 4-6, 6-4.
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Serena reaches Aussie quarters as opponent ill
Serena reaches Aussie quarters as opponent ill
