Kvitova lost to Anisimova in Indian Wells in March of last year and said she was well aware of her opponent's talent.
Kvitova: It's been a while since I made the second week
“I played much better than last year for sure," Kvitova said on Eurosport.
I knew it would be a tough match for me. Someone you have lost before it is always kind of difficult to play her again, but I started pretty well and nerves went a little bit out.
The two-time Grand Slam winner claimed the Sydney title earlier this month and hopes to return to her best level after a disappointing 2018.
Highlights: Kvitova crushes Anisimova
“I’m trying to continue my performance from Sydney and I feel good," she said. "It’s really important for me, I am really enjoying the time to be on the court and play tennis which is the main thing for me.
“It’s been a while to be in the second week and playing the quarter-final of a Grand Slam so I’m really enjoying it.”
Kvitova: I'm here to play - not think about world No. 1
Left-hander Kvitova booked her first quarter-final appearance in Melbourne since her run to the 2012 semi-finals and first at a Grand Slam since the 2017 U.S. Open.
She will face Ashleigh Barty in the quarter-finals.