City found a late equaliser from Bernardo Silva to cancel out Alejandro Garnacho's opener for Manchester United to take the match to penalties.
And after a tense shootout, City won 7-6 following Jonny Evans' skied effort for United.
Afterwards, Dias said: "Obviously it feels great to start the season winning. We have lost here in the last three years.
"We're happy to start the season in the way we have and obviously there's a lot to come. It's another trophy and we're always happy to collect trophies. We've just had a long season and we still don't have a full squad back but it is very good for us to start this way.
"It's very different circumstances to the FA Cup final but it's a final and we love to win finals. The way we played, conceding a goal but then to make it 1-1, I'm very proud of all of us and very happy with the win.
"Thank you for coming, this is very special for us and we know what this means to you. It's for them why we're here."