Gauff, whose lead-up to the year's final Grand Slam included hardcourt titles in Washington and Cincinnati, converted four of her five break point opportunities and surrendered her serve just once during the 76-minute match in Arthur Ashe Stadium.
Gauff said she took a more aggressive approach into her second meeting with Andreeva after having to rally back to beat the Russian in three sets at this year's French Open.
"She's the type of player if you give her something she's going to take advantage and today I think I did well making her play off her back foot," Gauff said during her on-court interview.
"But Mirra, I think she has a great future ahead of her, I've been in her position before ... she's going to be back on this stage many more times in the future."
For home favourite Gauff, it was a much more routine day at the office considering she had to dig deep to secure a comeback win in an opening match that left her frustrated with the amount of time her opponent, Laura Siegemund (35), took between points.
Andreeva has relished the Grand Slam stage in her debut season on tour having reached the third round at the French Open and fourth round at Wimbledon but was not able to conjure up similar magic in New York.
Gauff quickly settled into a rhythm after dropping serve in the opening game and during one stretch won 12 consecutive points to open up a 4-1 lead before relying on her serve to close out the set.
Gauff continued to show solid movement and shot selection in the second set where she feasted on Andreeva's serve, twice breaking the Russian to love to build a 5-2 lead before closing out the match with a hold to love.
Up next for Gauff will be a clash with Belgian 32nd seed Elise Mertens (27), who was a 3-6, 7-6, 6-1 winner over American Danielle Collins (29) in a match that lasted nearly three hours.
