The defending champion was joined by fourth-ranked Agnieszka Radwanska and former No. 1 Caroline Wozniacki. Radwanska beat Australian qualifier Anastasia Rodionova 6-3, 6-2, and Wozniacki downed Sorana Cirstea of Romania 7-6 (7), 6-0.
"Today was a tough match," Stephens said of the wind and 80-degree temperatures Fahrenheit. "The conditions were tough but Klara played well. Now I just need to regroup and keep working on my game and get ready for Dubai."
Stephens pulled out of Fed Cup against Italy last week after aggravating an abdominal muscle injury and suffering from a stomach virus.
Azarenka will retain her top ranking if she reaches the final and Williams fails to reach the semifinals. She made a nervous start, dropping serve to fall 2-0 behind. But the Belarusian recovered to break three times and win the first set. Azarenka then took a 3-0 lead in the second set.
"Well, it wasn't easy for sure, but you never want to take this as an excuse because it's the same for both players," Azarenka said of the wind. "I started to go too much for my shots at the beginning. Then I started to be more patient, work more the points."
Radwanska had little trouble with the 131st-ranked Rodionova, breaking to go up 3-1 on the way to winning the first set. The second set was even more one-sided, with the Pole breaking twice on the way to victory in which she said "the wind changed every couple of minutes. It was very tricky."
Radwanska has started the season well, winning in Sydney and Auckland and reaching the quarterfinals of the Australian Open. Having reached the Wimbledon final last year, she feels ready to challenge for the No. 1 ranking.
"It's always a little bit of pressure, but I think this is the dream for everyone to be in the final of a Grand Slam and, of course, to be No. 1 in the world," she said. "I was really close to that last year, and definitely I'm going to try to do it this year."
Several seeded players lost, including the 16th-seeded Stephens. Marion Bartoli, seeded ninth, lost to Svetlana Kuznetsova 6-4, 6-4, while No. 17 Lucie Safarova of the Czech Republic was beaten 6-4, 6-2 by Christina McHale of the United States. Urszula Radwanska, the sister of Agnieszka, upset 15th-seeded Roberta Vinci of Italy 6-2, 5-7, 6-0.
Wozniacki saved four set points in the tiebreaker before going up 1-0 when Cirstea hit a crosscourt backhand wide. The second set was a procession with the Dane breaking three times.
"In the second set, I came off really strongly and probably crushed the last of her beliefs in that match," Wozniacki said.