James McPake scored inside the opening minute for the Dark Blues before a goal from substitute Leigh Griffiths early in the second half earned the Hoops a draw.
The home side got off to a flying start when McPake met a driven corner from Philip Roberts and sent a header from 12 yards past Adam Matthews on the line with just 49 seconds on the clock.
Dundee were clearly buoyed by their bright start and Peter MacDonald was inches away from getting his toe on a waist-high delivery from Roberts as Celtic were almost caught out with a swift counter-attack on 16 minutes.
The visitors survived another scare six minutes later when Craig Gordon got down to divert a Roberts drive on to the base of the post after the Dundee winger cut inside Emilio Izaguirre with some ease.
Roberts then opened up the Celtic defence with a lofted pass from just inside his own half and although MacDonald beat the hesitant Gordon to the ball his flick landed just wide of the post.
Celtic had not been so comprehensively outplayed in a domestic fixture for some time, but they almost levelled the game seconds before the break when Efe Ambrose met a Tony Stokes free-kick on the six-yard line and his downward header was spectacularly turned over the bar by Kyle Letheren.
The visitors stared the second half far brighter and equalised on 55 minutes when Griffiths collected a pass from fellow substitute Eoghan O’Connell and turned at the edge of the box to dispatch a low shot which took a slight deflection off Thomas Konrad to send the ball into the far corner of the net.
Celtic then dominated possession, but were unable to cause Dundee too many problems.
Instead, Gordon (78) had his palms stung by Martin Boyle at his near post after the Dundee substitute had been released by a weighted Paul McGowan pass.
And when the pair combined again two minutes later Boyle sent a low drive past the far post when his better option was squaring to the unmarked Luka Tankulic in front of goal.
Dundee almost paid when Stokes smashed a shot against the outside of the post from 12 yards at the other end.
And a minute later Letheren had to be alert to keep out a vicious drive from Griffiths, but in the end a draw was a fair result.