Adams will join the Bundesliga club when the winter transfer window opens on January 1, and has signed a contract through the 2022-23 season.
The 19-year-old has emerged as one of the top young talents in the United States player pool, having become a key figure for the Red Bulls since making his MLS debut in 2016 as a 16-year-old.
Adams has been a regular starter for the Red Bulls the past two seasons, featuring both as a right back and as a central midfielder.
“We’re genuinely happy for Tyler and his family and we know that he is ready for the next step," Red Bulls head coach Chris Armas said in a team release.
"In these last four years, I've learned that he is simply incredible. Tyler has a maturity, fearlessness, work ethic, and desire to win that make him unique."
When Adams joins his new side next month, RB Leipzig manager Ralf Rangnick expects him to step in and play right away.
"We've been following Tyler Adams for a long time," Rangnick said in a club release. "At 19, he was a key player this season for the New York Red Bulls, and he made his senior international debut for the USA.
"He'll be an immediate option for the midfield, and we're sure he'll take the next step in his development in the Bundesliga."
Adams has earned nine caps for the United States after making his senior national team debut against Portugal in November 2017.
In Germany, the teenager is looking forward to following in the footsteps of two of his international teammates.
"If I think about the top leagues in the world, I think about the Bundesliga. A lot of youngsters have made the next step in their development here. RB Leipzig are all about that, so it's a great step for me," Adams said.
"For me, it's the chance to prove myself at the highest level and measure myself against my friends and compatriots, Christian Pulisic of Dortmund and Weston McKennie of Schalke."
At Leipzig, Adams will link up with former Red Bulls manager Jesse Marsch, who is now an assistant under Rangnick.