Up against 27th seed Anisimova early on day two, four-time grand slam champion Osaka never looked comfortable on her least favourite surface as she struggled to control her powerful game.
Anisimova, the 20-year-old who reached the semi-finals at Roland Garros in 2019, also beat Osaka in the third round at the Australian Open in January.
On Court Phillipe Chatrier, top-seeded Iga Swiatek extended her winning streak to 29 matches by advancing to the second round in Paris.
The Polish player, who won the French Open title in 2020, beat Lesia Tsurenko 6-2 6-0 in 54 minutes to begin her campaign.
Swiatek's winning streak is the longest by any woman since Serena Williams won 34 in a row in 2013.
She has won the title at her past five tournaments and has also won a tour-leading 14 sets by 6-0 this season.
Meantime the first three games on court Suzanner-Lenglen 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 on game point.
After a couple of match points went begging, 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.
With AP.
