The Fifers are currently languishing at the bottom of the Irn-Bru Second Division but have already claimed an SPL scalp in the shape of local rivals Dunfermline in the cup competition.
Brown said: "When you're bottom of the league as they are, it's going to be difficult to come from where they are and get promotion.
"So what is their objective? The two cups. I'm not discounting them from winning the league or anything but it's more realistic that they can have a good cup run.
"They've already had good success, the local team in Fife beating Dunfermline. If we are complacent at all, we could pay for it."
Aberdeen secured a 6-0 win over East Fife in the Scottish Cup last season but Brown believes that scoreline flattered the Pittodrie side.
He added: "We are well aware that we've got a game on here.
"I've had a look at our cup game against them last season. More than half of their team are still around.
"Although we won in terms of goals very comfortably, the game was much more even than the score suggests.
"I'm expecting it will be a tough tie. There is a capability about East Fife that we've got to be concerned about."