Each set took just 22 minutes as the veteran continued her quest with a stunning win over the 18-year-old.
"Today was a great day," said a beaming Serena after her match.
Playing a record 80th main draw match at Melbourne Park against an 18-year-old on her first trip Down Under, Williams was merciless under the lights of Rod Laver Arena, blasting 24 winners and winning all 10 points coming into the net.
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"I love your game and that forehand, it is really awesome,” said Serena after the match.
Kasatkina raised rambunctious cheers from sympathetic fans in the terraces when she prised a game in each set but Williams finished full of running, closing out the match in 44 minutes when her frazzled opponent pushed a forehand wide.
And while the scoreline was one-sided, Williams paid a nice tribute to her opponent at the net, telling her how much she liked her game, and then saying in her post-match interview what a "bright future" the young Russian has.
Williams, bidding for seventh title at Melbourne Park and a 22nd grand slam trophy, faces another unseeded Russian in Margarita Gasparyan in the next round.
Serena Williams v Darya Kasatkina: As it happened
Additional reporting via Reuters
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