James Chester and Curtis Davies found the net to put the Tigers 2-0 up inside eight minutes, but goals from Santi Cazorla and Laurent Koscielny forced the game to extra time.
Aaron Ramsey popped up in the box in the second additional period to complete the comeback and seal the Gunners' first piece of silverware for nine years, but Bruce remained proud in spite of the disappointment.
"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 build-up to Arsenal's two goals in normal time 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."