Brown spent four years as manager of Hull and guided them into the top flight for the first time in the club's history in 2008.
"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," Bruce told Sky Sports News.
"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."
Southend defeated Championship side Millwall 4-1 in the third round.