In her first match since her third-round exit at Roland Garros earlier this month, 21-year-old Osaka struggled with her first serve against the world number 33 and took two hours to dispatch her Greek opponent.
However, the Japanese, who lost to Sakkari prior to her U.S. Open title run last year, did manage to fire seven aces and won five break points to clinch victory on a surface she had previously admitted was not her strongest.
Elsewhere, former French Open champion Jelena Ostapenko stormed into the next round with a 6-0 6-2 demolition of Iga Swiatek while Petra Martic saw off the challenge of Ekaterina Alexandrova with a 6-3 7-5 win.
However, rain led organisers to bring an early end to the day's play.
Venus Williams's match against Aliaksandra Sasnovich and French Open champion and second seed Ashleigh Barty's encounter against Donna Vekic were both pushed back to Wednesday.
(Reporting by Rohith Nair in Bengaluru; Editing by Toby Davis)
