The draw means Swallows, who are currently MTN8 Cup holders, will not have the opportunity to defend their crown next season.
Mogorosi could have given Celtic the advantage in the fourth minute when he headed over the bar from close range at the back post after a sublime cross by Lyle Lakay.
Five minutes later Katlego Mashego was inches from giving the Birds the lead when his free-kick from just outside the centre the box brushed the top of the net.
Mogorosi had an opportunity to atone for his earlier miss when he unleashed a powerful shot from 20 yards, but Swallows keeper Greg Etafia was on hand to snuff out the shot.
With 11 minutes left in the first half, the home crowd was on its feet as Obada let rip with a powerful shot from just outside box which Celtic keeper Patrick Tignyemb could not stop.
It was a wonderfully taken shot after Mashego had done well to fend off the Celtic markers and tee up his teammate.
However, Celtic hit back soon after the restart and found the equaliser in a rather fortunate way. Lakay’s cross from the left was not dealt with by the Swallows defence and the ball bounced off Mogorosi’s chest and into the back of the net.
In the 65th minute an unmarked Lakay could have given his team the lead when he headed wide of goal, despite having all the time in the world to control and finish off.
In the dying minutes of the match Rodrick Mutuma was denied by the crossbar with Etafia completely beaten, but the match ended 1-1.