The Taysiders are well adrift at the bottom of the Scottish Premier League table and odds-on to return to the First Division.
But Smith saw enough in last week's 1-1 draw at Easter Road to give him further hope that his side can start narrowing the gap - starting against Kilmarnock.
"We definitely still have the belief," he said.
"We are totally focused on getting off the bottom of the league. The dressing room has the confidence - not arrogance - that we can do that.
"We took a lot of encouragement from last week. We were positive from the first minute.
"Easter Road is not an easy place to go but we put them under pressure, especially in the first half and we need to take that into our home games, again tomorrow.
"We have found out that mistakes are punished and if you are not consistent then you will get found out.
"We know what we have to do, we have to start picking up three points at home tomorrow.
"Kilmarnock have been up and down but they have dangerous players, like (Paul) Heffernan who is a real danger in attack.
"They like to get the ball down and play, so hopefully it will be an attractive game."
Smith added further cause for optimism by buying playmaker Gary Harkins from Kilmarnock yesterday, although he will not be able to face his former club as a condition of the deal.
Smith said: "We all know the quality he has and he is a great character to have on and off the pitch as well.
"He is once more showing his commitment to the cause here at Dens Park and I know he was loved by the Dundee fans during his last spell.
"I'm also very pleased to have had the backing I have received by the board of directors in our pursuit of Gary as it wasn't easy and other clubs also tried hard to secure his services."
Chief executive Scot Gardiner added: "We have been working on this deal tirelessly for weeks now in the face of fierce opposition and I'm over the moon to finally put things to bed.
"Gary is ecstatic to be joining us as am I and we are laying down a marker on where we want to take the club."
Harkins trained with his team-mates today and is looking forward to returning to a dark blue jersey against Hearts at Tynecastle on Wednesday.
"The deal took a while but I'm happy to get it done and I'm glad to be back," Harkins told DFCTV.
"I'm delighted to be back and just looking forward to playing in front of this crowd.
"It's still a good dressing room and it's good to come in and see the boys."