Former Italy striker Inzaghi, 40, has signed a two-year deal to step up from his role as a youth-team coach.
Seedorf, 38, replaced Massimiliano Allegri in January with the club struggling in 11th in Serie A. He won 11 of his 19 league matches in charge, but Milan finished eighth, 45 points behind champions Juventus.
The former Netherlands international played for Milan from 2002 to 2012, while former team-mate Inzaghi spent 11 seasons at the Rossoneri before retiring in 2012.
Inzaghi, a World Cup winner with Italy in 2006, helped Milan win Serie A titles and two Champions League crowns as a player. (BBC)