The Dutchman had already announced that he would take charge of the Serie A side on Tuesday and the Rossoneri have now formalised his appointment
Allegri was sacked on Monday, less than 24 hours after a shock 4-3 loss to Sassuolo that left the Rossoneri 11th in Serie A and 20 points adrift of the Champions League qualification places.
Former Milan midfielder Seedorf announced the following day that he had decided to end his playing career in order to return to San Siro as coach.
Mauro Tassotti took temporary charge for the midweek Coppa Italia clash with Spezia but Seedorf met with Milan vice-president Adriano Galliani on Thursday and put pen to paper on a contract that will run until June 2016.
The 37-year-old Dutchman will take charge of his first training session at 15:00CET, while first game at the helm will be Sunday's Serie A meeting with Verona at the Giuseppe Meazza.
Seedorf, who was capped 87 times by Netherlands, spent 10 years with Milan, with whom he won two of his four Champions League medals.