Neil Lennon knew his side could take a big step towards the league title after Rangers were held at home to Kilmarnock, while Dundee were looking put some distance between themselves and bottom-place St Mirren with Hamilton in 10th earning a crucial win themselves on Saturday against Hearts.
It appeared that the home side would earn a valuable point with an excellent rearguard performance, led by Genseric Kusunga. However, deep into injury time Odsonne Edouard netted to send Celtic fans into rapture and Lennon down the touchline in celebration.
Following the international break Celtic know they can all but confirm their eighth successive league title with a win over Rangers in the Old Firm derby at Parkhead.
It was the home side who would have the first effort of the game with midfielder Ethan Robson clipping a shot narrowly over the bar from the edge of the box after some good build-up play.
But that was as threatening as the home side would be for a while as Celtic took control and dominated proceedings.
In the 10th minute, Callum McGregor exchanged a one-two with Mikey Johnston but shot straight at Sonny Dieng before James Forrest blazed over the ball after the ball fell to him inside the box.
Then in the space of five minutes Sinclair was denied twice by Dieng, both following shots from the edge of the box. First McGregor's wayward effort was directed goalwards by the Englishman but the goalkeeper reacted smartly.
A few moments later, in the 20th minute, Scott Brown hit a daisy cutter straight at the Dundee stopper but he could only palm the ball into the path of the onrushing Sinclair but he once again reacted quickest to thwart the 29-year-old.
Dundee's defence was holding firm with plenty of bodies in and around the box, making it difficult for the away side but also to create chances of their own.
Celtic's best opportunity of the half fell to Sinclair, who felt he should have had a penalty earlier, again. This time Forrest picked Martin Woods pocket, slipped in his team-mate but his curling effort was pushed away impressively by Dieng.
On the stroke of half time Dundee should have taken the lead. Fine hold up play by Kenny MIller brought Robson into play, who fed Scott Wright breaking from midfield but Scott Bain in the Celtic goal was equal to his shot.
The second half was a slog for all viewers. Dundee rarely ventured into the Celtic half while Celtic huffed and puffed but were unable to blow the Dens defence down.
Johnston sent in a dangerous cross seconds after the restart but the visitors would have to wait until the 80th minute before they came close to opening the scoring.
Jonny Hayes, positive down the left after his introduction, sent in a cross which founds its way to the feet of Forrest. However, the in-form attacker could only turn and see his shot deflected just past the post.
Just when it appeared Celtic would be unable to penetrate the Dundee defence Hayes burst into the box and cut back for Edouard to find the back of the net.