Auger-Aliassime, who has struggled this season with a knee injury and at times apparent loss of confidence, suffered the first service break of the match in the fifth game and that was all the Italian would need to capture the first set.
But the Canadian turned the tables over the next two sets, never being broken again while finding the range with his forehand and moving well to extend points.
He beamed a broad smile when Berrettini's forehand went long on match point, grateful for the bright spot in a thus far disappointing 2023 campaign.
"My mindset today was no matter what happens, I wanted to have the best attitude possible, the best effort from the first point to the last," said Auger-Aliassime, who improved to 14-13 on the season.
"And I feel like I was able to turn that frustration from the first set - having break points and losing them, not being careful enough with my service game - and turning that into something positive to bounce back really strong.
"It has been a challenging season. I've been very fortunate, every year I have grown in the game and improved my ranking but this year it's a new challenge and I need to accept it as well."
The 12th-seeded Auger-Aliassime will look to keep his winning ways going in his first-ever meeting with Frenchman Mannarino in the second round.
Elsewhere, Australia's Jordan Thompson toppled American John Isner in a pair of tiebreaks for a 7-6(5) 7-6(2) win in a match where neither player broke serve and Isner crushed 20 aces.
Finland's Emil Ruusuvuori and Nicolas Jarry of Chile also advanced in the morning before evening rain halted the play of several other first-round matches at the U.S. Open tune-up event.
