The 21-year-old has signed a three-and-a-half-year deal with the Premier League side but will spend the rest of the season with Championship side Boro.
He left Manchester United's academy at 14 to study in the United States.
"I can't wait to get started over here," Harrison said. "I'm looking forward to testing myself in England."
He added: "Middlesbrough are a good club, with Premier League experience, and they're in the mix for a return via the play-offs this season."