The Dark Blues set up a quarter-final tie with city rivals Dundee United by winning 5-1 against the First Division league leaders, gaining some welcome respite from being bottom of the Scottish Premier League.
Smith's side sit 14 points adrift at the foot of the table after a strained first season back in the top flight following a seven-year absence.
However, their boss believes they can still defy the critics to stay up.
"I know what these guys are capable of." Smith said. "I've always had great faith in them and that will always be the case.
"Results haven't always gone according to plan this season but we've never lost hope at any stage.
"We will keep on focusing on one game at a time but hopefully we can kick on from here and take it into the league."
Dundee face Ross County when they return to SPL action on Friday night, and Smith reckons their emphatic victory over Morton in the cup can set them up perfectly to meet the Staggies.
He said: "We've got to look to go onwards and upwards.
"The lads are all in good spirits after this result and so they should be.
"There is plenty still to play for between now and the end of the season.
"It was great to see us score so many goals as well."