The appointment of the 50-year-old Maldini in the new role is part of Milan's attempt to restore the club to past glories having finished no higher than sixth in Serie A since 2013.
Maldini, whose number three shirt has been retired by the club, made a record 647 appearances in Italy's top flight, winning 26 trophies including seven Serie A titles and five European Cups.
He also played 126 times for his country, finishing as a runner-up at the 1994 World Cup and Euro 2000.
"There are few words to describe what Paolo Maldini represents for AC Milan," club chief executive Paolo Scaroni said in a statement on Sunday.
"Today's appointment is yet another sign of Elliott's commitment to build a strong foundation for long-term success.
"It will not be easy and it will take time, but we have ambitious objectives, and Paolo's appointment is an important step toward returning AC Milan to where it belongs."
Milan splashed out over 200 million euros ($231 million) on players last summer but finished sixth in Serie A and failed to qualify for the more lucrative Champions League.
The club was taken over by US hedge fund Elliott last month after the Chinese owners of the Italian giants failed to repay debt.
Milan signed Argentina striker Gonzalo Higuain from Juventus in a three-player swap deal on Thursday that saw Leonardo Bonucci return to Turin with Mattia Caldara moving in the other direction.