In a game that was far from memorable but nonetheless fiercely contested, the Birds had the better of the play, but lacked precision needed in front of goal.
Celtic, meanwhile, remain one of just three teams who are still unbeaten after eight rounds of the League. The others are Bidvest Wits and leaders Orlando Pirates.
Chances were few and far between in the first half, particularly for Celtic.
Swallows looked more fluid in the early stages of the second half, but some poor touches in the final third undid the good work.
On 55 minutes David Radebe did well to pull the ball back from near the by-line into the path of Lungisani Ndlela, but the lanky striker skewed his shot hopelessly wide.
Ndlela had no better aim with his head after meeting a corner-kick just past the hour-mark.
The home side looked the likelier scorers in the final phase of the match, but Celtic held them at bay with some ease.