Russia's Kasatkina, runner-up at the last two editions of the Australian Open warm-up event, broke Australia's Olivia Gadecki five times to advance 6-2, 6-3.
"I think Adelaide is the tournament I've played most in my career," said the 27-year-old, who made six WTA finals last year and won two titles.
"It's the beginning of the season, I'm happy to be back.
"My tactic today was to turn on my legs and win as many points as possible and play aggressively when I had the chance."
The match featured four service breaks in the first five games as third seed Kasatkina got re-acquainted with the Memorial Drive centre court.
She will next face last year's Australian Open and Wimbledon junior runner-up Emerson Jones, a wildcard entry who scored an impressive 6-4, 6-0 win over China's Wang Xinyu.
Croatia's Vekic, a Wimbledon semi-finalist last year, saw her Australian Open build-up go from bad to worse with a 6-2, 6-3 defeat to Kazakhstan's Yulia Putintseva.
The world number 19 was also beaten in two group matches in Perth at the United Cup.
Victory moved Putintseva into a second-round clash against either American fourth seed Danielle Collins or Tunisian wildcard Ons Jabeur, a three-time Grand Slam finalist.
"Today I played great, I was focused all the way through," said Putintseva.
"We've played so much and know each other's games well. There are no secrets on the court."
In other matches, 2021 US Open finalist Leylah Fernandez was eliminated by Russia's Ekaterina Alexandrova 6-4, 3-6, 6-2.
The first Grand Slam of the year begins in Melbourne on Sunday.
