The Tigers were 2-1 up at the Hawthorns until the Baggies scored twice in the last three minutes to progress with a 3-2 win.
In their last two Barclays Premier League matches Hull threw away leads against Newcastle and West Ham to draw both games 2-2 and it is becoming a concern for a side who prided themselves on defensive rigidity last season.
The arrival of Manchester City at the KC Stadium on Saturday is hardly the easiest opportunity to prove that the message is sinking in, with Manuel Pellegrini's side having ran in seven unanswered goals against Sheffield Wednesday in midweek, but Bruce has laid down the challenge.
"It's a concern, of course it is, and it's hugely disappointing," said Bruce.
"The things we've managed to do in the last couple of years, to get out of the Championship, stay up and get to the FA Cup final, have all been based around the ability to hang on to a lead, the ability to see it through.
"I can't remember in 16 years of management, or the 30-odd years I've been in football, where we've conceded such goals in the last couple of minutes to cost us.
"But it's our turn
At West Brom it's cost us our place in the cup.
"So we'll have to get back to basics, if that's what it is, the basics.
"We'll certainly have to defend properly on Saturday against a really outstanding team."
One positive from Hull's cup exit was the chance to see Hatem Ben Arfa on the pitch for a full 90 minutes.
The Frenchman was an outcast at Newcastle before sealing a loan move on transfer deadline day and Bruce has been cautious about over-exposing the playmaker.
A much-changed XI for the cup provided the chance to test his legs but Bruce believes more time is needed before he can be considered a regular starter.
"There's no question he can pick a pass, he's got fantastic ability, but we also have to bear in mind he hasn't really played since last February, which is a hell of a long time," he explained.
"He might have a part to play on Saturday but there's a lot to do with his levels of fitness and he's still got a lot to do before we let him loose on the Premier League.
"It would be wrong of me to ask him to play two (full) games in four days when he hasn't played one in seven months.
"We have to box clever with him
Everybody knows what ability the kid's got and he'll be a big asset I'm sure, but we have to get him to the right fitness levels."
Source : PA
Source: PA