Cibulkova dropped just two games in her 6-2 6-0 demolition of Germany's Laura Siegemund as the reigning WTA Finals champion set up a second-round meeting with Eugenie Bouchard of Canada.
Wozniacki survived a second set meltdown to beat Olympic champion Monica Puig 6-3 2-6 6-4.
"It's a tough match, because you kind of feel like 'oh, I'm winning pretty comfortably right now' but then she can go on a streak," said the Dane.
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"That's where it started going in the second set, and in the third set I tried to stay closer to the baseline and maybe play a little deeper," Wozniacki added.
Cibulkova needed three gruelling sets against Siegemund in their only previous meeting last year in Wuhan but was a lot more assertive on Monday.
"I just felt pretty good, you know, overall when I practise here, and of course match and practice is different story, but I was ready," said Cibulkova, who suffered a quarter-final defeat at the hands of Frenchwoman Alize Cornet in Brisbane last week.
"I knew it was going to be tough, but I had very good answers for her good shots. I was really solid, and when I had the chance, I was going for my shots.
"The chances I didn't take in Brisbane against Cornet, today I took almost all of them. So sometimes it's like this. I really appreciate this performance I had today."
Sixth-seeded Brit Johanna Konta and ninth seed Roberta Vinci also progressed with straight-sets victories.
Konta defeated local favourite Arina Rodionova 6-3 6-4 while Vinci beat Greek qualifier Maria Sakkari 6-4 6-0.
There was more heartbreak for local fans as Australian veteran Sam Stosur bowed out after her 6-3 6-1 defeat by Anastasia Pavlyuchenkova.
The defeat prolonged former U.S. Open champion Stosur's poor build-up for the Australian Open, starting in Melbourne next Monday, as she had also lost in the first round to Garbine Muguruza in Brisbane last week.
"I don't really know what to say. I don't feel like I played bad," Stosur said after the match. "I don't feel like I played obviously as well as I can."
(Reporting by Sudipto Ganguly in Mumbai; editing by Amlan Chakraborty)
