The 13th seed had lost to Davidovich Fokina on clay in Monte Carlo last year but found the right formula on New York's hard courts, sending down 17 aces, firing 48 winners to 27 and saving nine of 13 break points.
After a lacklustre first set in which Berrettini produced only six winners and failed to earn a break point, Italy's Australian Open semi-finalist came to life in the second set tiebreak and immediately broke his opponent in the third set.
Down a break in the final set, Davidovich Fokina suffered an injury in the sixth game when his leg slipped out from under him and he banged his fists on the court in pain.
He saw a physio and for a moment it seemed he might be done for the day but the Spaniard bravely returned to the court to finish the match to roars of encouragement from the crowd.
"I'm really proud because I didn't start the match the way I wanted to," said Berrettini, who has been sidelined a number of times this season due to injuries or illness.
"Obviously this is not the way I wanted to finish the match but I'm going to take the win."