The match eventually kicked off half an hour late to allow Celtic fans to make their way to the ground after a long delay en-route.
Dundee started brightly and forced a couple of corners early on but the visitors soon took control and their late arriving fans were all standing on their seats with seven minutes played when their team took the lead.
A free kick down the left by Steven Crainey saw Bobo Balde tower above the Dundee defence to send a great header into the far corner of the net.
Celtic were dealt a blow when Didier Agathe received a straight red card for a foul on Nacho Novo as the Dundee striker raced through on goal.
Steve Lovell should have done better for the home team when he met Novo's cross at the back post but the Englishman headed way off target.
Stanislav Varga fired a header just wide from an Alan Thompson corner a minute before half time.
With just over an hour of the match played Mo Sylla broke free on the right but his cut-back toward Henrik Larrson lacked accuracy and Dundee cleared the danger.
A good run down the left by Fabian Caballero saw him set up Novo at the back post but the little striker headed weakly past.
Moments later Julian Speroni got down low to save a drive by Sylla.
Dundee-born Mark Fotheringham, recently signed from Celtic, made his debut when he replaced Lovell with 13 minutes remaining.
With a minute to go, Dundee substitute Garry Brady blazed a shot high over the crossbar.
Dundee pushed everyone forward in the four additional minutes but could not find an equaliser.