Donnarumma shook off a hand injury and was given the armband as the Azzurris climbed to the top of their group with a 2-1 win over the Hungarians.
Goals from Nicolo Barella and Lorenzo Pellegrini rendered Gianluca Mancini's second half own goal a mere consolation.
European champions Italy have turned to youth after failing to qualify for a second consecutive FIFA World Cup and Donnarumma is glad to have led the new crop of players coming through.
"We knew that we had to change and restart with young players after a difficult period," Donnarumma said.
"They [the youngsters] came here with the right spirit knowing that you have to do everything you can for this shirt.
"The captain's armband is an indescribable emotion, wearing the number one shirt and the armband is something that leaves you speechless.
"For this shirt, I would play even without a finger, I have to thank the medical staff who helped me a lot; with my desire and their help, I was able to play tonight.
"For me, this shirt is incredible and I will play as long as I can with this shirt."
Head coach Roberto Mancini added: "We played a good game, especially in the first half.”
"Their goal, however, created us some fear and pressure in a game we should have won with a bigger margin of goals. Both teams were a bit tired at the end.
"We are playing with several youngsters and we are aware that we have plenty of work to do, the road is still long.
"We can go forward with confidence, however, because we played two good games against difficult teams in Germany and Hungary."
Italy are next in action against England on Saturday.
