The Dees have not won in the Scottish Premier League since November 17 and find themselves 15 points off second-bottom St Mirren.
Dundee chief Scot Gardiner has now begun the search for a replacement for Smith who secured just three victories in 26 outings for the Dens Park outfit.
“The board hope changing manager can bring about a change of fortunes,” said Gardiner.
“We want to stay in the SPL and stay in the Scottish Cup.
“Nobody at Dundee is giving up on staying up or even getting to the Cup Final. Both those things are important to us, we are not calling it a day.
“We have already had people in touch about the job. In three hours we have been swamped with potential candidates.
“I have now got the task of identifying someone to take charge of the team.”