Trailing 1-4 in the opening set, the pony-tailed Belarusian then gave herself a wake-up call and seized control by winning 11 of the next 12 games to improve her record this season to 15-0.
Azarenka clinched the first set with an ace, then broke Hantuchova for a sixth time in the seventh game of the second when the twice former champion dumped a backhand into the net.
"First round matches are never easy and Daniela is such a great player," Azarenka said courtside at the Indian Wells Tennis Garden. "She started really well. It was my first match after a while, so it's always a little bit tricky.
"She already had a match under her belt, plus I think she really feels very good here, winning two times before. So it caught me a little bit off guard. I needed a bit of time to realise ... that I had to step it up if I want to win the match.
"I adjusted really well, tried to make her hit out of her comfort zone, because at the beginning she was really going for her shots and making incredible winners. I changed the momentum."
Azarenka, who demolished Russian Maria Sharapova 6-2 6-3 in last year's final at Indian Wells, will next face Belgian Kirsten Flipkens, who earlier scraped past Romania's Monica Niculescu 4-6 6-4 6-3.
In other matches, former U.S. Open champion Samantha Stosur eased into the third round with a commanding 6-3 6-4 victory over American teenager Madison Keys while fourth-seeded German Angelique Kerber swept past Romania's Irina Begu 6-3 6-2.
Eighth seed Caroline Wozniacki of Denmark battled past Frenchwomen Alize Cornet 6-4 3-6 6-3 and 2008 champion Ana Ivanovic of Serbia hammered American wildcard Taylor Townsend 6-1 6-2.
Australian Stosur, seeded seventh in the elite WTA event at Indian Wells, broke her opponent's serve once in each set to win the first match of the day on the showpiece Stadium Court after an hour-and-a-quarter.
In bright desert sunshine, Stosur sealed victory on her first match point when wildcard Keys netted a forehand from the baseline and will next meet China's Peng Shuai, a 6-0 4-6 6-3 winner over Romania's Alexandra Dulgheru.
"I'm happy with the way I played," Stosur, who won her only grand slam singles title at the 2011 U.S. Open, said. "I thought I served very well.
"As soon as I put a little bit of pressure on her early on in her service game ... I really took a bit of a stranglehold on the match."
Stosur was lavish in her praise of the 18-year-old American, who reached her first WTA quarter-final in Sydney earlier this year.
"She's got a big game," the Australian world number nine said of Keys. "Her forehand is massive and she's got a very big serve. She's just got to put all the pieces together.
"She's only 18. I think she's got a lot of potential. She has those big weapons that win these big matches."
In earlier matches, 10th-seeded Russian Nadia Petrova eased past Swiss Stefanie Voegele 6-2 6-3 and Belgian Yanina Wickmayer beat Croatia's Mirjana Lucic-Baroni 3-6 6-3 6-2.