Osaka served for the set at 5-3 but fluffed her lines. Tig failed to take advantage and handed over the opener.
During the second set, the 26-year-old Romanian dug in. But serving to stay in the match at 5-6 down, she coughed up a match point.
A rasping forehand winner down the line saved that and she came through to take it to a tiebreak.
In the shoot-out, Tig looked dead and buried when Osaka raced into a 5-2 lead. But two sloppy shots brought it back to 5-4.
Tig could not capitalise and Osaka surged again to claim the tie in one hour and 47 minutes.
Snub
In the prelude to the tournament, Osaka declared on social media that she would not go to her post-match press conference at the French Open.
The 23-year-old from Japan said she found that questions from reporters were often intrusive and opened up doubts.
Gilles Moretton, the head of the French Tennis Federation - which organsies the French Open - described her move as an error.
She faces a fine of up to $20,000 for skipping mandatory news conferences.
Osaka answered questions on court after accepting a bouquet of flowers from the federation for her mother to mark Mother's Day.
“Thank you for coming,” said Osaka to the handful of spectators. “Happy Mother's Day to all mothers in the crowd. And I hope everyone enjoys watching the matches here today.”
Osaka, the winner of four Grand Slam tournaments, has never progressed further than the third round in her four previous visits to the French Open.
She will next take on another Romanian, Ana Bogdan, who knocked out Italian lucky loser Elisabetta Cocciaretto 6-1, 6-3.
