Wozniacki silenced the critics when she claimed her first Grand Slam title earlier this year at the Australian Open, becoming the first Danish woman to win a major.
The 27-year-old world No. 2 Wozniacki moves on to the fourth round where she will face either reigning US Open champion Sloane Stephens or 20th seed Daria Kasatkina of Russia.
"I managed to get my feet going more, and to start playing more steady, and that paid off today," said Wozniacki.
Wozniacki challenged a call in the final game when the line judge ruled that Sasnovich's shot caught the line. Wozniacki asked for a second look and the review showed that the line judge mistakenly called the ball in.
That levelled the score at 30-30 and Wozniacki won two out of the next three points she needed to take the third set and the match in the $16 million joint WTA/ATP hardcourt tournament.
Wozniacki hit two aces and won 64 percent of her first-serve points in the two hour, 17 minute match.
Wozniacki said the court conditions were difficult but she managed to make the adjustments.
"These courts are really difficult to play on. That's also why you see a lot of upsets," she said.
"The ball bounces really high and it goes extremely slow.
"Those two things are making it trickier to play. But at the same time, I'm just trying to adapt and trying to do my best out there and trying to move well."
Sasnovich, of Belarus, committed 14 double faults and had her serve broken seven times.
- Musical chairs for No. 1 -
Wozniacki had been one of seven women who held the No. 1 ranking who hadn't won a Grand Slam when she triumphed in Melbourne in January. The victory also allowed her to reclaim the world No. 1 ranking for the first time in six years.
She and Simona Halep have taken turns being No. 1 this year with Halep currently on top.
If Wozniacki wins in Indian Wells she will reclaim it but she says she is not thinking about it.
"Winning the Australian Open and getting the No. 1 ranking back at that point was extremely important to me and something that I'm very proud of," she said.
"Right now, that No. 1 ranking isn't what's on the forefront of my mind. It's really just trying to win every match I'm playing and trying to win this tournament."
In other women's third round matches on Monday, French seventh seed Caroline Garcia defeated Australia's Daria Gavrilova 7-5, 6-4.
Garcia is trying to improve on her best result here which is reaching the round of 16 in 2015 and last year.
"I know against Daria it's always going to be a fight until the end and she always wants to come back in the match so it's never over," said Garcia. "I'm very happy with the way I played."
Danielle Collins also won her third round match, defeating Sofya Zhuk of Russia 6-4, 6-4.
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