Up against 27th seed Anisimova, four-time grand slam champion Osaka never looked comfortable on her least favourite surface as she struggled to control her powerful game.
The first three games in Paris on Monday all went against serve in a tense start but Anisimova held to go 3-1 ahead.
A backhand error by Anisimova allowed Osaka to level for 3-3, but a double fault in the 11th game allowed the American to move 6-5 up and she held serve to love to take the set.
Anisimova broke decisively for 4-3 in the second set as Osaka again double faulted game.
After a couple of match points went begging, 20-year-old Anisimova finished it off in one hour 30 minutes with a stunning backhand winner down the line.
Osaka withdrew from the tournament in 2021 after boycotting post-match media duties for mental health reasons.
In the build-up to last year's French Open, Osaka said she would not attend the obligatory news conferences for players after matches as they impacted her mental wellbeing.
The Japanese, who at that time revealed she had been suffering from depression for almost three years, returned to competitive tennis at the Tokyo Olympics after the saga cast a spotlight on players' post-match media duties.
Roland Garros organisers conceded they could "do better" on mental health this year.
