The Tigers found themselves two goals up inside 15 minutes but conceded to Chris Smalling and Wayne Rooney prior to the half-hour mark.
James Chester, who had earlier netted for Hull, then headed into his own net to hand United the win leaving Bruce to rue some defensive errors made by his side.
"We pride ourselves on not giving anything away and unfortunately today we made too many mistakes," he told Sky Sports.
"We made elementary mistakes and then all of a sudden it isn't your day, when it could have been because we had the opportunities and attempts.
"We got off to a great start but we just needed then to keep the ball better. All credit, though, we stuck at it and on another day we might have got the result.
"We've had three or four opportunities to get the next goal in front and they are the defining moments in the game, and unfortunately they've gone down the other end and we've made a mistake and they've punished us.
"Against all the big teams we've played, we've been like that, and that's the way we'll continue."
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