Salzburg paid an initial $6 million for the 19-year-old and will pay a further $3 million if certain incentives are met, according to ESPN's Jeff Carlisle.
The Union will also receive a percentage of any future transfer fee Salzburg receive for Aaronson.
"This has always been the goal of the club. We want to establish a unique identity of developing young players through our academy with an end goal of cultivating and developing their talent to the level that allows them to compete against the very best in the world," said Philadelphia sporting director Ernst Tanner.
"We are incredibly proud that Brenden is leading the way as the first Philadelphia Union homegrown player to test himself against top European competition."
Aaronson, who has made one appearance for the United States men's national team, will join a club renowned for producing elite talent. Over the past seven seasons, Salzburg have sanctioned the sales of Sadio Mane, Erling Haaland, Dayot Upamecano, Konrad Laimer, Naby Keita, Takumi Minamino, and many other Champions League regulars.
Aaronson has started 18 MLS matches in the 2020 campaign, scoring four goals and assisting three times.