The 21-year-old USA international will move to Elland Road for an undisclosed fee on a five-year contract, running until June 2027.
Aaronson has been a standout performer for the Austrian champions this season, scoring six goals and registering 10 assists in 41 appearances across all competitions.
The former Philadelphia Union midfielder moved to Salzburg in January 2021 and linked up with current Leeds boss Jesse Marsch, who was in charge of the Austrian outfit between 2019 and 2021.
Speaking on Aaronson's arrival, Marsch told the club's official website: "Brenden is an incredibly hard worker and we're pleased to welcome him to the club.
"He adapts very well to tactical ideas and levels of play and his mentality is to do whatever it takes to help the team succeed. I know he will fit in here perfectly with our players and club with this mindset."
Leeds' director of football Victor Orta added: "We're pleased to have reached this agreement, we've been really impressed with the talent and skill set of Brenden, along with all the physical data collected on him.
"We believe he will improve the squad and we are really happy, it is very difficult to find a young player of this level, especially with Champions League experience. He can play in a number of positions and we believe he is the perfect player fit for us.
"We've been interested in Brenden for a long time and with taking the decision to hire Jesse Marsch and their experience of working together in the past, we wanted to close the deal as soon as possible which we now have done."
Aaronson is now set to join up with the USA national team ahead of their friendly encounters against Morocco, Uruguay, Grenada and El Salvador next month. body check tags ::