Bruce and Brown are the only two men to have managed Hull in the top flight, with Brown taking the Tigers into the Premier League in 2008 before being forced to drop down to the fourth tier with the Shrimpers.
And Bruce says his side must work extra hard to make sure they are not the victims of a cup shock at Roots Hall on Saturday.
"It's got a little edge to it, with the former manager being the manager of Southend and, of course he'll be desperate to turn us over which is normal for a smaller club in a lower division," said Bruce.
"It's got the classic symptoms (of a cup upset), so we've got to make sure we treat it like we did against Middlesbrough last time and produce a performance to get us through to the next round.
"Phil's done a terrific job down there, got them knocking around at the top end of the division and you can't ask for any more than that.
"What would put the icing on the cake for him is to turn us over and we've got to guard against that and make sure it doesn't happen.
"All the cameras, all the people who go to the game and all the people in the country will want the underdogs to win, which is normal.
"We've seen over the years all the shocks and surprises, that's what makes the competition what it is, but we've got to make sure we're not one of those shocks and surprises."