After just two minutes, the first chance fell to Celtic when Lassad Nououi played in Gary Hooper, who fired his shot just over.
Charlie Mulgrew curled a 30-yard free kick just wide of the post after five minutes, as Celtic had the better of the opening stage.
Dundee's appeals for a penalty were turned down after eight minutes when Emilio Izaguirre appeared to bring down Gary Irvine in the area.
After 16 minutes, Dundee nearly grabbed themselves a goal against the run of play. Colin Nish found himself in on goal, but he could not beat Fraser Forster, who saved well.
Some intricate play from Hooper and Lassad nearly resulted in a goal for Celtic on 19 minutes, but Hooper's shot was saved by Rab Douglas.
With 37 minutes gone, Celtic won a penalty after Kyle Benedictus brought down Lassad. Scott Brown stepped up to take the spot kick, but Douglas was equal to it and pushed the ball wide to keep the score at 0-0.
The Bhoys increased the pressure and made it 1-0 just before half-time through an emphatic finish from Hooper.
Celtic came out as the better side once again in the second half and nearly doubled their lead through a defensive mix-up. Irvine attempted to chest the ball back to his goalkeeper and ended up hitting the post before the ball was cleared.
Two minutes later, Celtic did get a second goal when Victor Wanyama smashed his shot off the post and into the back of the net from distance.
Hooper thought he had scored his second and Celtic's third after 58 minutes, but his goal was ruled out for an offside.
A minute later, Lassad got on the end of a cross from Adam Matthews, but his header was impressively saved by Douglas.
Some great work from Tony Watt on the byline resulted in a cross, which was met by Hooper, though the forward could only hit the post.
The final whistle blew and Celtic deservedly took all three points.