The Tigers put in a sterling display at Wembley and raced into a two-goal lead after just nine minutes as James Chester and then Curtis Davies struck.
Santi Cazorla's free-kick on 17 minutes brought Arsenal back into the game and they took the match into extra-time when Laurent Koscielny converted from close-range after 71 minutes.
Then Hull's hearts were broken with Aaron Ramsey firing home a very good finish with just 11 minutes of extra-time remaining.
"On another day we could have won it but in the end we didn't have enough," said Bruce.
"We are totally disappointed because it could have been one of those memorable FA Cup wins where the underdog goes and wins it.
"In terms of effort, endeavour and determination you couldn't fault them to a man. Proud? Of course I am. They were magnificent."
Meanwhile, Bruce's son, Alex believes that just being in the FA Cup was a reflection of how far Hull have come in recent seasons.
"A lot of the players were playing in the Championship this time last year," he told Sky Sports.
"To come into the Premier League, stay there, and get into an FA Cup final, the lads deserve a massive pat on the back."
Asked if his side were daring to dream at 2-0 up, Bruce replied: "We were, but you always think to yourself 'have we gone 2-0 up too early?'
"Everyone knows in the game if you sit off Arsenal and camp in your own half, with the players they've got, they can hurt you. Ultimately, that was the price we had to pay.
"We can take massive confidence out of that game today. If we can take this into next season, kick on next year and hopefully have a taste of days like this to come, it'll be fantastic."