Two-time champion Maria Sharapova had a tougher time subduing British qualifier Heather Watson 6-7 (7), 6-3, 6-4, while 2011 Wimbledon champion Petra Kvitova, the fourth seed, lost to Petra Martic of Croatia 6-4, 6-4.
Other winners included Grand Slam champions Li Na and Sam Stosur.
Azarenka played her first match since losing a thriller in the U.S. Open final to Serena Williams.
"This was my first match after a little break and that's never easy,'' she said. "But I was able to play some good tennis and I enjoyed it.''
Next up will be Roberta Vinci of Italy, who beat qualifier Pauline Parmentier of France 6-3, 6-1.
Watson pushed Sharapova to three sets in their last match at the 2011 U.S. Open, and did it again by winning a marathon tiebreaker in the first set.
"I was facing someone who had gone through (qualifying) and already played so I was definitely the rustier of the two,'' said Sharapova, who had 14 double faults. "Overall, it was just one of those matches where you are glad to be through.''
The second-seeded Sharapova, who won here in 2005 and 2009, took control of the third set when she broke Watson's serve for a 4-3 lead. She sealed the win when Watson sent a forehand return long.
Sharapova's next opponent, Lucie Safarova of the Czech Republic, moved on when compatriot Klara Zakopalova was unable to play because of a left wrist injury.
"She's a tough player,'' Sharapova said. "She had a break today with the walkover while I played three hours but hopefully that won't be a factor.''
Kvitova said she had trouble establishing her game against the hard-hitting Martic, who was playing her second tournament since recovering from a foot injury at the French Open. This was the biggest win of her career.
"Petra has really improved her forehand,'' Kvitova said. "She hits it very hard down the line and crosscourt and in addition to her serve has a good slice which can be tough for players to handle.''
Li defeated Anastasia Pavlyuchenkova of Russia 7-5, 4-6, 6-2, and Stosur overcame Francesca Schiavone 4-6, 6-2, 7-5.
Also advancing were Angelique Kerber of Germany, Sara Errani of Italy, Marion Bartoli of France and Nadia Petrova of Russia.