The 19-year-old Peru international joined Bayern's academy seven years ago from Augsburg's youth setup and spent the second half of last season on loan at 1. FC Köln, where he made just seven appearances without a goal or assist as he searched for regular game time.
Bayern Munich have also extended his contract to 2028 as part of finalising the Magdeburg move, a sign the club still consider him a prospect worth retaining long-term interest in, even while looking to accelerate his development away from the Allianz Arena.
Chávez said he was looking forward to the fresh start in Saxony-Anhalt.
"1. FC Magdeburg offers young players very good opportunities to develop further and take the next step.
"That's why I'm really pleased about the chance to gain regular game time at a high level here and keep improving.
"Now I want to get out on the pitch with the team as soon as possible, get to know my new teammates and settle in on the Elbe," Chávez said of his move.
Bayern's sporting director Christoph Freund explained the thinking behind the move, noting that Chávez had used his time at Köln to take further steps in his development and should now continue that progress with regular football at Magdeburg.
Bayern had fielded interest from several other clubs before settling on Magdeburg, including fellow 2. Bundesliga sides Eintracht Braunschweig and VfL Wolfsburg, newly promoted SC Paderborn, and Portuguese top-flight club Estrela Amadora, who have recently taken on former Bayern figures Thomas Müller and Mats Hummels as investors.
Chávez impressed in Bayern's pre-season, scoring in a recent friendly against 1. FC Heidenheim, and will now look to translate that form into consistent minutes in Germany's second tier, with a first home appearance for Magdeburg possible as soon as this weekend against Holstein Kiel.