He failed to save any of the penalties while Italy’s Pio Esposito and Bryan Cristante missed theirs.
“I cried because of the enormous sadness I’m feeling, along with the entire Azzurri team, of which I’m proud to be captain, and I know you, fans of our national team, are feeling it too,” Donnarumma added.
The Manchester City keeper, who was handed his senior debut in 2016 aged 17, is yet to play at a World Cup and will have to wait until he is at least 31 for his next opportunity.
He was 15 when Italy lost against Uruguay and exited the 2014 World Cup in the group stage.
He wrote it would take “a lot of strength, passion, and conviction” to restore his country “back where it deserves to be”.
The former Paris St-Germain and AC Milan keeper added: “Always believe; this is the driving force behind moving forward. Because life knows how to reward those who give their all, without holding back.
“And this is where we must start again. Together. Once again.”