City took a two-goal lead into the break at Wembley but shipped three in the second half to allow their local neighbours to lift the first trophy of the season.
Kompany told Sky Sports News: "We were a bit tired. Pre-season has been long and heavy and you are looking to be at your best at the beginning of the season and we will be.
"As I said the result could have gone either way. We have shown we are not normally the sort of side who concedes goals once we are all set."
The winning goal was handed to Nani after a mistake from Kompany himself allowed the Portuguese winger to go one-on-one with Joe Hart in the last minute.
"It happens, if you are a centre-half with a lot of responsibility but you have to take it - that is what football is all about," Kompany admitted.