Scott Robertson's fierce right-foot shot from 20 yards was held at the second attempt by Peter Cherrie on five minutes.
Good play by Colin Marshall set up Simon Lynch 12 yards out but his shot crashed against the bar and Garry Brady fired the rebound well over on 27 minutes.
Paul McHale saw his low drive from 20 yards go inches wide of the target on the stroke of half time.
Steven Craig bundled the ball into the net early in the second half but the flag was up for offside.
A powerful 30-yard free-kick from Paul Dixon was saved by Cherrie in the Clyde goal on 58 minutes and moments later Stephen O'Donnell headed poorly straight at Dundee keeper Scott Murray.
Clyde struck on 64 minutes when Dougie Imrie's cross from wide on the right took a wicked deflection and sailed high over Murray and bounced into the far corner of the net.
A fine save by Cherrie denied Steven Craig the leveller on 68 minutes.
Kevin McDonald should have doubled Clyde's lead on 73 minutes but he shot straight at Murray from eight yards.
Robert Mann sent a header just wide and Steven Robb's shot was saved in the closing minutes and when the final whistle sounded what was left of the home crowd could barely muster a clap between them.