Osaka admitted in her pre-tournament press conference that she still does not feel comfortable on grass.
However, she got off to a flying start in her first match since losing in the third round of the French Open.
Osaka broke the Sakkari serve three times in romping through the first set.
The world number 33 fought back in the second to take the match to a deciding set.
Osaka needed to stave off five break points in the fifth game to take a 3-2 lead but then pounced immediately for the only break in the final set.
Yulia Putintseva will face Osaka in round two after easing past Harriet Dart 6-1, 6-4.
Former French Open champion Jelena Ostapenko set up a second round clash with Britain's Johanna Konta with a 6-0, 6-2 win over Iga Swiatek.
World number three Karolina Pliskova will face her twin sister Kristyna Pliskova in the next round after the latter beat Viktoriya Tomova 6-3, 6-4.