The Dark Blues' faithful have had to endure a tough season in the Clydesdale Bank Premier League, and head into this weekend's cup fifth-round tie bottom of the table.
Smith hopes to give the supporters something to cheer about as they prepare to take on the Irn-Bru First Division leaders, but says they will have to play their part.
"They can be so important to us getting through," Smith said.
"It has not been easy for anyone this season, but this is a break from the league. We have to try to seize the chance and, when the fans are with you, it is so much better. It gives the whole place a lift and we want to get a result for them."
Dundee have collected only 14 points from their 24 SPL games so far, and lie 14 points adrift of second-bottom St Mirren. But Smith and his team will look upon tomorrow's game as a chance to gain some fresh impetus.
"These are always tricky ties, so we have to approach it properly," he said.
"We will be fully focused and look to get the kind of result which will send everyone home happy."
Dundee could hand a debut to on-loan Nottingham Forest midfielder David Morgan tomorrow, but midfielder Gary Harkins is cup-tied following his recent move from Kilmarnock.
The hosts are still without Stephen O'Donnell, Davide Grassi, Carl Finnegan, Mark Kerr (all knee) and Jamie McCluskey (Achilles).