The Italian international was a part of the cup-winning squad, making a 66th-minute appearance in the extra-time 2-1 win over Crystal Palace at Wembley on Saturday.
"We are very proud for the staff, for my team-mates and for the fans because we won this trophy and we fought until the end to win it," Darmian told manutd.com.
"I think this kind of match is the best to win because we came back after their goal and after Chris (Smalling) received a red card. We kept fighting and, at the end, won it with a fantastic shot from Jesse [Lingard]. It was a great goal to win us the game.
"It's always very good to win silverware and we are very happy.
"I hope it is the first of many. It is my first season and I think it was always going to be difficult to change culture and also the football but I tried to adapt quickly. I think it was a good season for me as, at the end, we won this trophy."