Lundmark beat Steve Mason with a wrist shot in the fourth round to nail down the fourth win in the last five games for the Northwest Division leaders.
Columbus lost its fifth consecutive game.
Curtis McElhinney made 29 saves in goal for Calgary. Mason made 29 saves.
Calgary had to battle back with two goals in the third period to tie the score. Dawes forced overtime at 15:32 of the third with a one-timer on an assist from Dave Moss behind the net.
Notable deaths of 2014 [PHOTOS]
Arizona Cardinals player formally charged