James Chester and Curtis Davies found the net to put Hull 2-0 up inside eight minutes, but goals from Santi Cazorla and Laurent Koscielny forced the game to extra time.
And Aaron Ramsey popped up in the box in the second additional period to complete the comeback and seal Arsenal's first piece of silverware for nine years.
Despite the defeat, Bruce was quick to praise his side's performance.
"We struggled at the death, but in terms of their effort, endeavour and determination, you couldn't fault them to a man," he told ITV. "In the end, we just didn't have enough.
"Proud? Of course I am. They were totally magnificent."
Bruce also stated his belief that refereeing decisions in the lead-up to Arsenal's two regular-time goals cost his side.
"I didn't think the first one was a free-kick," he said. "Certainly the goal they scored from the corner, we think it's a goal-kick.
"But I don't think it's the time now to whinge.
"We're totally disappointed because we had a wonderful opportunity and it could have been one of those wonderful moments when the underdog goes and wins the FA Cup.
"On another day, we could have won it, but in the end we just didn't have enough."
