Donnarumma, who made his debut for Milan as a 16-year-old and is already regarded as one of Italy's top goalkeepers, stunned the supporters in June when his agent Mino Raiola told the club he did not want a new deal.
But, after a month of negotiations, Milan said on Tuesday that Donnarumma would stay at the club until 2021.
"I was raised at this club and I never had any doubts in my head about staying," Donnarumma told reporters. "I'm very happy and proud to be with Milan."
"I say sorry to the fans, I repeat that I have never had the intention of betraying them and I thank them for all the support they have given me in these two seasons."
Donnarumma made his senior debut in 2015 under coach Sinisa Mihajlovic and quickly established himself as the first-choice goalkeeper.
He has already made 68 Serie A appearances and became the youngest goalkeeper to play for Italy when he made his debut as a 17-year-old against France last year.
He was pelted with fake cash by supporters during a match for Italy against Denmark at the European under-21 championship in Poland last month.
On Wednesday, Milan announced the signing of Donnarumma's older brother Antonio, also a goalkeeper.
The 27-year-old has not enjoyed the same success as his brother, however.
He was raised at Milan but never appeared for the first team. Instead, he went on to play for Piacenza, Gubbio and Bari in Serie B.
He also made one Serie A appearance for Genoa - his only game in Italy's top flight to date - and spent last season at Greek club Asteras Tripolis.
(Writing by Brian Homewood in Berne; Editing by Alison Williams)
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