"I'm just trying to do the best that I can and obviously. try and play my best tennis", Barty, who is assured of keeping the top ranking through to the end of the Australian Open, said after her Brisbane exit on Thursday.
Barty, who will begin in the second round at Memorial Drive after a bye, takes on Russia's Anastasia Pavlyuchenkova, who defeated Estonian Anett Kontaveit 6-2, 6-4.
"It was a really entertaining match for us and I could feel that the people enjoyed it a lot", she said.
She survived a first set against Kasatkina featuring five breaks of serve and secured her second-round place with a break in the final game of the second.
Barty had said she expected to quickly pick up the pieces of her game after her Brisbane setback.
Barty denied beginning the season as the top-ranked player in front of her home fans had put extra pressure on. Every match I play here will be a bonus, I've done my preparation well.
He was the second seed to pull out of the ATP-WTA event after Novak Djokovic opted to head instead to Melbourne to prepare for his title defence at the opening Grand Slam of the year after leading Serbia to the ATP Cup title in Sydney.
Nearly a year after he lost to Nadal at the Australian Open (winning just seven games in a three set loss) his game has come on immensely, he said.
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Mirza and Kichenok will now face Vania King and Christina McHale of United States for a place in the semi-finals. It was a testing opener that went to the match tiebreak after both pairs had one set each.
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"This is my first win of the year, I'm happy about that".
The top seed is slated to run into former world No.1 and three-time grand slam champion Angelique Kerber, 2018 US Open victor Sloane Stephens, a qualifier or talented Czech Marketa Vondrousova in the quarter-finals.
Barty also fell to a qualifier in Brisbane, with world No.53 Jennifer Brady stunning the hometown favourite at Pat Rafter Arena in straight sets.
Yet Vondrousova will have her hands full with Rodionova, who before defeating Stephens upset Carla Suarez Navarro and Anastasia Potapova to reach the main draw. "I crave it. I love it", she said. It's always great to get such wonderful wins against the top players.
Elsewhere, Maria Sakkari of Greece made quick work of Vitalia Diatchenko of Russian Federation 7-5, 6-2.
The first Grand Slam of the season opens next week in Melbourne.
Earlier, American qualifier Tommy Paul beat Slovenia's Aljaz Bedene 6-4 6-4 and Brit Daniel Evans accounted for Kazakhstan's Alexander Bublik 7-5 6-2.