With temperatures forecast to top 102 degrees, Azarenka didn't want to be out in the heat.
"I felt like I'm back into the competitive mode," she said.
The No. 94-ranked Daniilidou only won 10 points in the first set and, despite having triple break point in the fourth game of the second, couldn't win a game in the second set.
Azarenka's win came a day after No. 2-ranked Maria Sharapova, the woman she beat in the Australian final last year, recorded her second 6-0, 6-0 win of the tournament.
The top-ranked Azarenka had her friend RedFoo in the stands watching and signing autographs, and said she went onto the court listening to a "great mix of disco music and a little bit of new music. I really start to like it a lot -- there's no words really."
Azarenka won her first Grand Slam title here last year, during a 26-match winning streak to start the season.
She didn't win another major, despite being close to victory in the U.S. Open final against Serena Williams, but held the No. 1 ranking for most of the season.
"It's pretty difficult to duplicate something like that," she said, "all I can do is try."
The temperature hit 93 degrees during her match, the first of the day on the center court, and was forecast to rise above 102 degrees during the day.
Serena Williams was playing the next match at Rod Laver Arena, hoping to get through the second round against Garbine Muguruza on her tender right ankle.
Williams hurt the already heavily taped ankle in her opening 6-0, 6-0 win on Tuesday. After winning Wimbledon and the U.S. Open last year, it seems injury is the biggest threat to Williams winning a third consecutive major title.
Sharapova was careful not to show any disrespect to the two women she beat 6-0, 6-0 along the way to a third-round match against Venus Williams.
The No. 2-ranked Sharapova overwhelmed Japan's Misaki Doi in 47 minutes Wednesday, even less time than she'd needed to beat fellow Russian Olga Puchkova two days earlier in her first competitive match of 2013.
No woman had posted back-to-back 6-0, 6-0 wins at a Grand Slam since 1985.
Yet it didn't excite the 25-year-old Sharapova, who wasn't even alive when Wendy Turnbull did it at the Australian Open.
"It's not really the statistic I want to be known for," Sharapova said. "I want to be known for winning Grand Slams."
To Sharapova, her first two wins have been a matter of "getting the job done." She was recovering from a sore right collarbone that ruled her out of a tuneup event in Brisbane earlier this month.
She skipped the Brisbane tournament last year, also due to injury, and went on to reach the Australian Open final.
Sharapova leads the head-to-head series 4-3 against Venus Williams, but Williams has won both of their Grand Slam meetings.
Numbers don't mean much to Williams, who is making trip No. "Lucky 13" to Melbourne Park.
She's still on the comeback trail after missing seven months after the 2011 U.S. Open to deal with Sjogren's Syndrome.
Now seeded 25th, she dropped only one game in her first-round win and then went down a break early to Alize Cornet of France before winning 6-3, 6-3.
The biggest upset on day three was 2011 U.S. Open champion Samantha Stosur's 6-4, 1-6, 7-5 loss to China's Zheng Jie.
Stosur led 5-2 in the third and wasted two chances to serve out the match, finally giving it away with a double-fault on match point.
"It was a bit of a choke," Stosur said. "Obviously it's a hard one to take when you get yourself into a winning position and you lose five games straight."
No. 4 Agnieszka Radwanksa, 2011 French Open champion Li Na, No. 5 Angelique Kerber and 17-year-old American wild card Madison Keys all advanced in straight sets.
On the men's side, No. 1-ranked Novak Djokovic is already into the third round, extending his winning streak at Melbourne Park to 16 with a 6-1, 6-2, 6-3 win over 20-year-old American Ryan Harrison.
He's on track for a record third consecutive Australian Open title and doesn't have any other major winners on his half of the draw after Rafael Nadal withdrew and Federer, Andy Murray and Juan Martin del Potro landed on the opposite side.
Fourth-ranked David Ferrer beat American Tim Smyczek 6-0, 7-5, 4-6, 6-3. No. 5 Tomas Berdych defeated Guillaume Rufin of France 6-2, 6-2, 6-4, and No. 8 Janko Tipsarevic took almost four hours to oust Lukas Lacko 6-3, 6-4, 3-6, 4-6, 7-5.
No. 20 Sam Querrey moved on when fellow American Brian Baker retired in the second set with an injured right knee and was pushed off Court 6 in a wheelchair.
"He's the last person that deserves anything like that with his five or six surgeries already," Querrey said.