Linette needed two hours, eight minutes to oust Kuznetsova in the final tune-up before the Australian Open, which begins Jan. 20.
In other action, No. 5 seed Veronika Kudermetova of Russia eliminated Australian Samantha Stosur 6-2, 6-2.
Viktoria Kuzmova of Slovakia, Kristyna Pliskova of the Czech Republic and American Lauren Davis also all moved on with three-set victories.
Australian Ajla Tomljanovic survived a marathon opening set, fighting off two set points to defeat qualifier Yulia Putintseva, 7-6 (4), 6-2 in Adelaide, Australia.
Tomljanovic also had to recover from a break down in the first set to beat the persistent Putintseva, of Kazakhstan, in a match that lasted one hour, 58 minutes. She next will face Wimbledon champion Simona Halep, ranked No. 4 in the world and the tournament’s second seed.
Other first-round winners included No. 4 seed Belinda Bencic of Switzerland, No. 7 seed Sofia Kenin and No. 9 seed Angelique Kerber of Germany.
–Field Level Media