The Dens Park side are 15 points adrift at the foot of the Scottish Premier League table but the new interim manager has not given up hope of a remarkable escape job just yet.
And for that reason he has pleaded with the Dark Blue faithful to turn out in numbers for Wednesday night's home meeting with St Mirren.
Brown, whose side lost to city rivals Dundee United in the quarter-finals of the Scottish Cup at the weekend, said: "I think the fans appreciated what the guys put in against United as far as effort goes.
"They all gave everything and when they do that I'll never criticise them.
"We need the supporters to keep right behind us and hopefully they do that.
"We have to be attack-minded without going gung-ho.
"We know wins are ideally what we need to try to make up some ground on the teams above us.
"That's our job, to close that gap, and nobody here will ever be giving up on that."
The compact nature of the SPL table this season means just 16 points currently separate 10 of the 12 teams.
And for that reason, St Mirren boss Danny Lennon has not given up hope that his team, who sit in 10th place at present, could go even further than their original target of sixth.
A win tonight would leave them just seven points behind fourth-placed Ross County.
He said: "This game is of huge importance to us because we still have a strong desire to keep chapping at the door for the top six.
"This is our game over all the other teams in the league and I believe we are still capable of fighting for three of the top-six slots.
"Only Celtic, Motherwell and Inverness are guaranteed to be in the top six and the rest is up for grabs.
"There are plenty of twists and turns and a win would certainly keep us in there with a realistic shout.
"But it's totally down to us and we need to get the wins on the board."
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