The Dark Blues are currently 14 points adrift at the foot of the Premier League but produced a performance belying their position.
And the reward for Smith's men is a quarter-final tie with neighbours Dundee United, who also had an easy victory over lower league opposition on Saturday.
"It's a very good win all things considered," said Smith. "It should give the players a lot of confidence for the rest of the season.
"The goals all came from different players and there was a lot of decent football on show so it's all positive.
"We will hopefully take encouragement from this and there's no reason why we can't.
"This was always going to be a dangerous tie against a good side but we handled it well and I'm very happy with that side of things."
Morton boss Allan Moore admitted the First Division team were always struggling in the Dens Pak encounter.
Moore said: "It just wasn't good enough and we can't make any excuses. At the end of the day we didn't do the basics right, especially in defence."