The Dark Blues host First Division Morton at Dens Park on Sunday with a place in the quarter-finals up for grabs.
Smith's side sit 14 points adrift at the foot of the Clydesdale Bank Premier League following their midweek defeat by Hearts at Tynecastle and look certain to be relegated at the end of the season.
But Smith is urging his players to take advantage of their cup weekend in a bid to kickstart their season.
"It's been difficult this season for obvious reasons but this is a wee break from the league," he said. "The cup is always nice to look forward to and we're going into Sunday's game looking to try and get a run.
"We've found results hard to come by in the league but this is a one-off tie with a last-eight place going to the winners.
"We don't need any extra motivation than that so we'll definitely be ready for it. The lads are all excited about the prospect of this tie because we know what's at stake for the club.
"I won't need to say too much to them before the match. We want to get into the next round and believe we can do so if we play the way we know we can."
Dundee slumped to a disappointing 1-0 loss away to Hearts on Wednesday and Smith wants to see an instant improvement against Morton, who are currently five points clear at the top of the Irn-Bru First Division.
He said: "We could have done better the other night. We had played well in the two previous league games but didn't really carry that on in the Hearts game, altough I don't want to be too critical of the players.
"However, the good thing is that we have a game coming up very quickly where we can hopefully make amends.
"We've had Morton watched so will give them our full respect. They are doing well in the First Division at the moment and are obviously doing a lot of things right.
"They will come here full of confidence but it's up to us to take the game to them on the day as we're the home team.
"Hopefully the fans will get behind us as they always do and make a great atmosphere. They would love to see us getting a run in the cup and so would the players.
"It's all on the day when it comes to cup competition so we just want to get ourselves into the hat for the next round come Sunday night."
Dundee could hand a debut to on-loan Nottingham Forest midfielder David Morgan, but midfielder Gary Harkins is cup-tied following his recent move from Kilmarnock.
The hosts are still without injured players such as skipper Stephen O'Donnell (knee), Davide Grassi (knee), Carl Finnegan (knee), Mark Kerr (knee) and Jamie McCluskey (Achilles).