The midfielder has joined the Premier League club for a fee believed to be $29.5 million, sources have told ESPN, and on a contract until 2027.
"It's an amazing feeling to be here at this historic club, I am really excited for the new season," Aaronson said.
"I think it was around Christmas or January that I could see there was some interest and hearing about Leeds United being interested, it was a huge moment for me and I was super excited.
"I had known a lot about Leeds at that time and what a famous club it is, the fans they have and how dedicated they are, it was just a dream place for me.
"We're going to have a big next season and I am really excited to be a part of it. I know the team is going to be up for it, so it is going to be an important season.
"I am not even thinking about my off-season, I am just excited to get back into it!"
✍️ #LUFC is delighted to announce an agreement with Red Bull Salzburg for the transfer of Brenden Aaronson, which will be completed on July 1st— Leeds United (@LUFC) May 26, 2022
Aaronson becomes Leeds' first signing of the summer after they secured their Premier League survival on the final day of the season.
The transfer to Leeds sees Aaronson reunite with Jesse Marsch, who managed him at Salzburg in the 2020-21 season.
"Jesse is a fantastic coach," Aaronson added. "I worked with him for half-a-year at RB Salzburg and he was amazing. I was grateful to work with him then and I am grateful to work with him again, so I am excited.
"With the half-year I played under him, he taught me a lot. He taught me how to get into the final third to score goals and get assists, so that's something I am looking to bring here.
"I've heard about what the fans like about players here and I feel like I can be that type of player. I am a hard worker, I have this creative side to me and I think I fit this Leeds recipe in a player.
"The way Jesse wants to play, I learned a lot about the system and flourished under that system, so I am really excited to get back into it again."
Leeds were interested in Aaronson during the January transfer window, but he stayed at the Austrian club to play in the Champions League knockout stages while helping Salzburg to the Austrian Bundesliga and Austrian Cup domestic double.
Aaronson, 21, is regarded as one of the top U.S. prospects and looks set to feature for the USMNT at Qatar 2022. He has 18 caps, having scored five goals for the national team, and can play as a No. 10 or as an inverted winger.
He becomes Marsch's first signing after the American took charge of Leeds in February. Leeds were in the midst of a relegation battle and managed to secure their Premier League status on the final day of the campaign after they beat Brentford 2-1 as Burnley dropped down into the Championship after losing at home to Newcastle United.
The transfer will see Aaronson's first club -- Philadelphia Union -- receive in the region of $5m from the deal, according to ESPN sources. Aaronson excelled in Major League Soccer for the Union, making the MLS Best XI for the 2020 campaign. He then signed for Salzburg for €6m in January 2021 and has since won two Austrian Bundesliga titles and two Austrian Cups, while scoring 13 goals in 66 appearances.