Gauff, who won the US Open last year to break her Grand Slam duck after losing the 2022 final in Paris, needed less than an hour to advance, even though the match was briefly halted after Avdeeva accidentally bumped her head against the umpire's when she was contesting a point.
"Today was kind of a straightforward match. I didn't have to do much. I wouldn't say I played amazing. I just had to play solid today. That's what I did," Gauff told reporters.
Adveeda was making her Grand Slam debut in Paris and Gauff said she sensed her opponent's stress on the second showcourt.
"I do think she was a little bit nervous with the bigger court, but I played well too," Gauff said.
In the second round, world number three American Gauff will take on Slovenia's Tamara Zidansek who had reached the semi-finals in the French Open in 2021 and beat Belgium's Alison Van Uytvanck 6-2, 2-6, 6-1 on Sunday.
"She's definitely a fighter. I think she has, you know, good ball tolerance, she can make a lot of balls in the court, pretty good mover. So it's going to be a tough match," Gauff said.