Clark, 18, will remain on loan at the Red Bulls until the end of the Major League Soccer season before making the move to Germany, where he's signed a contract until 2024.
Sources told ESPN that Barcelona are among the sides that have been following Clark in MLS this season, with the Minnesota-born midfielder a product of the Barcelona Residency Academy in Arizona.
However, Leipzig, who will be coached by American Jesse Marsch next season, moved quickest to secure the signing of Clark.
"In Clark, we have once again pulled off the signing of highly talented young player and have been able to tie him down for the long term," Leipzig's commercial director Florian Scholz said in a statement.
"This signing completes another piece to our puzzle for our squad planning at an early stage. We decided that a loan back would allow Caden, at just 18, a whole season in the MLS before he joins us here at Leipzig."
Clark, who has not yet been capped by the United States, follows in the footsteps of Tyler Adams, who also joined Leipzig from the Red Bulls in 2019.
After beginning his career with Minnesota Thunder, Clark moved to the Barca Academy in 2017, where he spent three years.
In 2020, he was picked up by New York Red Bull's reserve side but was promoted to the first team after just 12 appearances in the USL Championship.
Prior to that, the Red Bulls had to acquire his MLS rights, which were held by Minnesota United, in exchange for $75,000 of general allocation money.
In 15 MLS regular-season appearances, he has six goals and two assists, including a team-leading four goals this season.
"Caden is a big talent and we are happy to see him take the next step in his career," Red Bulls head coach Gerhard Struber said. "He is a hard-working kid and has a bright future ahead of himself."