Seedorf took charge at the San Siro in January following the dismissal of Massimiliano Allegri and was under contract with the Rossoneri for a further two years.
The 38-year-old, who had no prior coaching experience, fell out with the Milan hierarchy in recent months.
A statement from the club read: "AC Milan announces it has dismissed coach Clarence Seedorf and has handed the reins of the first team to Filippo Inzaghi until June 30, 2016."
Under Seedorf, Milan won seven of their last nine matches but finished a disappointing eighth in Serie A and failed to qualify for European football for the first time in 16 years.
Inzaghi recently revealed he was ready to coach the senior team.
The 40-year-old spent 11 seasons playing for the club, where he won two Serie A titles and two UEFA Champions League trophies, before retiring in 2012.
Inzaghi, who is the club's top scorer in European competition with 43 goals, has been in charge of Milan's youth team for the past two seasons.
His performances at the helm of the youth team impressed Milan owner Silvio Berlusconi.
When asked about Inzaghi last week as a possible replacement for Seedorf, Berlusconi said: "I see Inzaghi very determined.
"He is hungry for victories.
"He is consistent with what we want Milan to be, that is a team that goes out to win and wins convincingly.
"A team that wins fair and that makes the fans happy with their performances."
Inzaghi has yet to be unveiled to the media but everyone knows how he feels.
"Milan is my life and my story," Inzaghi told AC Milan's TV channel last week. "If I get some good news (regarding coaching the first team) then that's great. Otherwise, I am proud to continue coaching these boys."
It remains to be seen what personnel Inzaghi will have at his disposal next season.
There is speculation that midfielder Kaka will leave, with the Brazilian tempted to move to Major League Soccer.
The Rossoneri have signed former Paris St Germain defender Alex and are also expected to acquire winger Jeremy Menez from the French side on a free transfer.
Inzaghi will no doubt be keeping close tabs on Mattia De Sciglio, Ignazio Abate and Mario Balotelli - the three Milan players that have made Cesare Prandelli's Italy World Cup squad.
However, Inzaghi will not have club captain Riccardo Montolivo available for the start of the season with the experienced midfielder having fractured his shin in Italy's recent international friendly against the Republic of Ireland.