Former world number one Caroline Wozniacki also went through as the leading players made short work of lower-ranked opponents on a subdued center court.
Seven of the top 10 remain in contention but Belarussian Azarenka decided to rest her injured right foot.
"I have been playing a lot of tennis over the last five days, so going to a new tournament is too much to handle right now - it keeps getting worse," said the world number two.
Australian seventh seed Stosur beat Russia's Ekaterina Makarova 6-3 6-1 after suffering one wobble on the way when she lost her serve to fall 3-2 behind in the first set after slugging an easy cross-court smash wide.
However, that shocked the 2011 U.S. champion into action and Stosur then won 10 out of 11 games as her opponent wilted.
Makarova, 24, scuffed a backhand into the net to lose the first set, while the second was a formality for Stosur.
"The first couple of games were good - I was getting ahead in the games but not winning them - and then I got down that break and that kicked me into gear," Stosur told reporters.
"It wasn't like anything outstanding, but I played tight games and didn't really give her too much."
Czech Petra Kvitova put an indifferent start to 2013 behind her as she thumped Daniela Hantuchova 6-2 6-1.
Kvitova won Wimbledon in 2011, but suffered a second-round exit at January's Australian Open.
"At the beginning of the season I wish (I had) better results," Kvitova told reporters. "I changed my fitness preparation, so I needed some time to get used to it."
Eighth seed Wozniacki was similarly dominant in beating Lucie Safarova 6-2 6-2 although the Dane lost her serve at 5-1 in the first set before breaking back immediately.
Safarova double-faulted to gift the world number 10 a 3-1 lead in the second set and the Czech capped a miserable display with a shanked forehand to cede the match to Wozniacki.
"I'm really enjoying the tennis and am happy to be out there - at one point it was just a job but I feel like I've got that hunger back and that enjoyment of the game," said Wozniacki.
The Dane ended 2010 and 2011 as the top-ranked women's player, but endured a tough 2012, slipping down the standings as she made little impression at the grand slams.
This year has also proved difficult for the 22-year-old, who has gone beyond the last 16 in only one tournament, and she will now face Chinese qualifier Jie Zheng before a potential showdown with new world number one Serena Williams in the quarter-finals.
Williams, who got a bye into the second round, will play France's Marion Bartoli on Wednesday and the American's chances of lifting a 48th singles tour title were boosted by the absence of Azarenka, who also pulled out of last year's Dubai event.
The Belarussian has been in fine form this season with a near-perfect record, winning 12 matches in a row since withdrawing from the Brisbane International semi-final against Williams in January due to "a bad pedicure experience".
That run includes winning her second Australian Open title, plus Sunday's Qatar Open final in which she beat Williams.
(Reporting by Matt Smith; Editing by Ken Ferris)