Hull went in at half-time 2-1 down after goals from Jose Baxter and Stefan Scougall had come either side of Yannick Sagbo's equaliser.
Bruce made a double change at the break, bringing on Matty Fryatt and Sone Aluko, and it was the former who levelled the scores four minutes after the interval before further goals from Tom Huddlestone and another substitute, Stephen Quinn, put Hull firmly in control.
Although Jamie Murphy pulled one back late on, David Meyler added a fifth on the break as Hull reached the first FA Cup final in their history.
"I'm absolutely delighted," Bruce told BT Sport. "It needed a hell of a turnaround.
"You have to give credit to Sheffield United, they took care of the ball better than we did.
"It was an awful first 45 minutes, we had to do something about it and thankfully we got it right in the second half.
"A lot has been written about the name change (proposed by club owner Aseem Allem) which has probably overshadowed the players' performances, but I'm delighted for the owners and everybody concerned."
Captain Curtis Davies, meanwhile, said reaching the final was a dream come true, regardless of their showing on the day.
"We've prided ourselves this season on our defensive performances and that wasn't one of our better ones," he said.
"Fair play to Sheffield United - they were much better than us in the first half, but we got the win and that's all that matters.
"It's what we dream of when we were kids. Today we all feel like fans when we celebrate and go to the final."
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