The fifth-round replay at the KC Stadium was far from a classic, but first-half efforts from Curtis Davies and Robert Koren were enough to see the Tigers into the last eight despite late nerves following Leonardo Ulloa's 68th-minute header.
Hull now host Bruce's previous club Sunderland, with a Wembley trip at stake and the East Yorkshire side having already done the Barclays Premier League double over the Wearsiders.
Bruce has pedigree in the competition, having won it with Manchester United, but Hull have only appeared in the quarters five times in their history and reached the semis on a solitary occasion 84 years ago.
"The last time was 1930, wasn't it? There's not many around who went to that one," Bruce said.
"We've given ourselves a wonderful chance. It's a nice position to be in and now we've got to see if we can make it a really memorable season.
"(Wembley) is a wonderful occasion and, when you sit back in your cardigan and slippers, you remember the FA Cup. I've been fortunate enough to win it and it would be nice to go back there again if we possibly could.
"It's all to play for. We've got to beat Sunderland yet but (Wembley) is a wonderful day out - the best day you'll have as a player or a coach. But more importantly it's for the supporters - if they can get that it's terrific."
Bruce would have been forgiven for thinking Davies' looping far-post header on the quarter-hour and Koren's deflected free-kick 20 minutes later had killed the tie before half-time.
In front of less than 11,000 spectators, Brighton seemed to be going through the motions but sparked into life after the interval.
Lewis Dunk hit the crossbar with a powerful strike before Ulloa planted a faint header past Steve Harper with 22 minutes remaining.
Hull's fluency deserted them as they ground out the closing stages of the game, but Bruce was not surprised to see the Sky Bet Championship side make a go of it.
"In the first half we were completely in control of it. We just needed a third and the game was over," he said.
"It was almost job done, but how often do we see that? Just at the right time Brighton got the goal and fair play to them.
"All of a sudden it was a cup tie again."
Seagulls assistant manager Nathan Jones cut a dejected figure after the game, bemoaning a missed opportunity and blaming a slow start.
"We didn't come out of the blocks in the first half. We didn't play with any real intensity," he said.
"We didn't show up. That wasn't an Albion performance in the first half. It felt like a bit of an anti-climax, the way we played, especially after what we'd done in the first game against them.
"We made a game of it and got the goal, but it was a mountain to climb against a good Premier League side."