On a baking Court Philippe-Chatrier, Gauff struggled to find her range early on and was a point away from slipping two breaks of serve behind against her compatriot.
But the fourth seed quickly recovered her poise to take control and move safely through to the second round.
Gauff double-faulted on set point at 5-3 in the opener but broke the Townsend serve in the next game to move ahead.
After that it was one-way traffic as she raced to victory.
"I have so many great memories on this court," Gauff, who beat Aryna Sabalenka in the final last year, said.
"I was a little nervous going out today, that showed a bit, but I was able to find my calm."