The Japanese third seed, 23, ground down Czech world number 87 Marie Bouzkova for a 6-4 6-1 first-round win.
She will play world number 145 Olga Danilovic of Serbia next.
Osaka withdrew from Roland Garros in May, revealing she had "suffered long bouts of depression" since winning her first major title at the 2018 US Open.
The four-time major champion decided not to play at Wimbledon in June but returned for the Tokyo Olympics, where she lost in the third round.
Bouzkova showed impressive early resistance in New York, but Osaka eventually got a break at 5-4 up to seal the first set.
The 2021 Australian Open winner came into the tournament having won only three matches in the past three months but looked to have recovered her form as she overpowered and outmanoeuvred her opponent in the second set.
After her win, Osaka said she felt "really comfortable" playing on Arthur Ashe Stadium again, adding it was "kind of crazy" to play in front of fans after the tournament was held behind closed doors in 2020 because of the coronavirus pandemic.
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