Horton, 20, a USA Under-23 international, has signed a two-year contract and will provide competition for the first team.
“I know the manager and assistant manager of the USA team and they gave him a glowing reference,” Rosler told the club’s offical website. “We’re very happy with him.
“He has fantastic physical attributes and is a really positive character. He really wants to learn.
“He worked with (Leeds’ new goalkeeping coach) Richard Hartis at Cardiff and we appreciate the fact that we can work with him again – that was a big factor in him coming to join us.”
London-born Horton, who moved to America at the age of nine, turned down a soccer scholarship at North Carolina University in favour of a move to Peterborough on the recommendation of former Tottenham goalkeeper Brad Friedel.
Horton spent a year at Peterborough before signing for Cardiff last summer. He worked closely there with Hartis, who also arrived from Cardiff on Tuesday and played a key role in his move to Elland Road.
Horton added: “I’m really excited.
“Any time you can come to a club like Leeds, where all the players are hungry for success and hungry to move up the table and back to where the club belongs in the Premier League, that’s every player’s dream.
“To be a part of that is something I’d love to have on my resume.
“I’ve heard nothing but good things about Leeds United as a club.”