The Steelmen hit the front in just the first minute through John Sutton but the Terrors managed to restore parity when David Goodwillie scored five minutes before half-time with a superb overhead kick.
Sutton grabbed a second for his side in the 72nd minute but the home side clawed back again almost instantly through Jon Daly to set up a replay, which has been scheduled for March 29 at Fir Park.
Dundee boss Peter Houston was full of praise for the efforts of Goodwillie who he believes sparked his side into action after a tardy start.
"He has come out with a sublime overhead kick. It was fantastic technique,” he said.
"Wayne Rooney took a lot of credit for that and Goodwillie has got to do the same.
"We were right behind it, it was a hopeful cross into the box, I'm thinking 'what are you going to do, are you going to take it down?' All of a sudden I see him taking off.