Murray, faced with the world record-breaking serve of the physically imposing Sam Groth, had all the answers as he won 6-0, 6-4, 6-1 in just 91 minutes against the flummoxed Australian.
Earlier, Ivanovic went through 6-3, 6-3 against Anastasija Sevastova in a match which was held up for 30 minutes when a fan was badly injured by a fall on some steps inside Rod Laver Arena.
"I felt really bad. I was really shaking, because the sound of the lady falling was really loud. I could hear it," said Ivanovic, who looked on anxiously as the patron received treatment.
"I think I'm feeling in the best shape body-wise, spirit-wise, everything-wise," said Azarenka, who has recovered from injury and a bout of depression following her split with musician Redfoo.
Progression was less smooth for world number three Garbine Muguruza, who struggled through an error-strewn first set against Kirsten Flipkens before finding her touch and winning 6-4, 6-2.
"I'm extremely happy because Kirsten is a very difficult opponent who has a unique game," said the 22-year-old Spaniard. "Every match is very tough and I am just happy to be here."
One significant match statistic was that Murray served more aces than 67th-ranked Groth 10-6, while breaking the Australian's powerful serve seven times.
"It's really impossible to say if it's the best (I've) played," Murray said, adding: "It's been a very good start, for sure. But I can still get better, I think."
Johanna Konta joined her fellow Briton in the third round when she beat China's Zheng Saisai 6-2, 6-3 -- only the third time she has made it so far at a Grand Slam tournament.
But there was heartbreak for Serbian former world number one Jelena Jankovic, the 19th seed who went crashing out 3-6, 7-6 (7/5), 6-4 to 97th-ranked German Laura Siegemund.
And Milos Raonic beat Tommy Robredo in unusual fashion when he secured victory with the only service break of the match -- after nearly three hours of play, and in the very last game.
In the women's draw, Britain's Johanna Konta maintained
"Against someone like Caroline, she's not going to give it to you. You really do have to earn it and win it till the very last point. I'm just very happy I was able to keep that pressure on," said the ninth seed.
Konta's prize is a round-of-16 rematch with Russia's Ekaterina Makarova, whom she edged at the same stage last year, going 8-6 in the final set.
Makarova, seeded 30, beat sixth seed Dominika Cibulkova in a three set marathon.
Mirjana Lucic-Baroni, who hadn't won an Australian Open match since 1998 before this year, extended her fairytale run by beating Greece's Maria Sakkari 3-6, 6-2, 6-3.
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