Kyle Benedictus, Steven Milne and Kevin McBride netted for the Dark Blues while Leigh Griffiths scored a last-minute consolation for the visitors from Edinburgh.
Dundee have now taken seven points from their last three matches, although they remain at the base of the SPL table - but just one point adrift of St Mirren.
Smith said: "It's a great result and a very good performance.
"We deserved to win the game and we got three goals as well which is always nice.
"We played with high energy levels and pressed them high up the pitch which we got a lot of joy from.
"We're pleased with the overall performance and the lads have all been working ever so hard.
"There's a long way to go but we'll concentrate on what we're doing. It's as simple as that. We're not looking at anyone else and that won't change."