The 18-year-old winger joined the Red Devils from Penarol on deadline day, making a €10 million (£9m/$12m) move from the Uruguayan club.
Pellistri admitted that making the move to Old Trafford is hard to wrap his head around, having only made his senior debut with Penarol in the fall of 2019.
"It's amazing that I only made my debut with the senior team a little more than a year ago," Pellistri told Telemundo.
"It was unimaginable and to play in the first team with Penarol and now to do it with Manchester United, I would never have dreamed of it and it is something very emotional for me and my family."
The winger admitted that he had an offer from Lyon in addition to Man Utd, but the draw of playing in a league he frequently watches on TV was too much to pass up.
"There was a lot of talk about the two offers and luckily they tried to leave me out of it as much as possible so as not to go crazy," Pellistri added.
"First came Lyon and then Manchester and I am very happy with the opportunity.
"It's a league that I always watch with my parents, seeing the great teams and you can't imagine going to play there.
"Now that I have the chance to play there, I'm eager to start a new adventure and see how the rhythm is and if it is as fast as they say."
Pellistri also expressed his gratitude for the club he is leaving behind, having advanced through the youth ranks at Penarol. Though he is only just departing, he admitted he is already thinking about coming back some day.
"It is the biggest team," Pellistri said of the Uruguayan giants. "I played for all of the club's youth teams, I was at the club for more than seven years. I lived everything, many experiences, very nice things like debuting in the Campeon del Siglo stadium among other things. It was a dream and a pleasure to defend the team.
"I am leaving with a lot of desire to return one day."