The 20-year-old midfielder joined the Serie A giants after failing to command a regular first-team place in United's line-up under Sir Alex Ferguson.
Pogba has revealed he decided to leave United after seeing Rafael and Ji Sung Park get the nod ahead of him in midfield for a Premier League clash with Blackburn Rovers.
Wayne Rooney had been dropped on disciplinary grounds, while Michael Carrick was forced to play in central defence, but Pogba only made the bench.
Pogba recalled the moment in an interview with French TV station TF1, saying: "I said to Ferguson: 'Play me, and I will show you if I'm ready or not'.
"Against Blackburn, on December 31, 2011, I was on the bench. Ferguson puts Rafael in midfield with Ji Sung Park. Then I gave up. I was really bummed."
Evra has since revealed his regret at United's failure to keep hold of his France team-mate, with both players in the French squad for upcoming games against Australia and Finland.
Pogba joined Juventus on a four-year contract after running down his United deal - and has made eight appearances so far this season, helping the Bianconeri up to third in the Serie A standings.
He recently bagged the winning goal in a 1-0 victory over Torino, and Evra admits his departure was a major blow to the defending champions.
"His [Pogba's] talent on the pitch, his aggression, his presence, is very rare," said Evra in an interview with TF1. "He very quickly earned the respect of players like Ryan Giggs and Paul Scholes.
"It's true that have you have to win over the manager too, but to have won the dressing room like he did [was impressive].
"It was truly a huge loss when he left the club."