Japhet Zwane had the first goal opportunity of the match in the seventh minute when he lined up a shot from just inside the big box, but although he struck the shot well, the ball whistled past the upright with Cosmos keeper Avril Phali struggling to make up the ground.
Three minutes later Phali was called into action when he did very well to keep out James Chamanga's shot with his legs after the Zambian striker had struck with a close range effort from Zwane's nicely-weighted cross.
Swallows midfielder Maimane Phiri seemed somewhat fortunate to escape with just a yellow card for what looked like a deliberate elbow in the face of Bamuza Sono in what was a bruising match.
Birds keeper Greg Etafia had little to do in the first half, but did well in the 16th minute to sneak the ball off the head of Sono.
The remainder of the half was a scrappy affair as the tackles flew in from both sides and there was little in the way of enterprising play as the game turned into a midfield battle.
Ezenkosi looked sharp just after the restart and had two chances in quick succession as first Clint Sephadi missed the target with a long range shot and then Nkosinathi Nhleko failed to hit the target with a header from a Dikgang Mabalane cross.
Two chances then followed in quick succession for the Beautiful Birds as Zwane missed with an effort from the left-hand side of goal and then Chamanga failed to profit from Cecil Oerson's delivery when he headed wide of goal.
Shots were then traded with Cosmos' Reneilwe Letsholonyane having an ambitious long range shot saved by Etafia, while for Swallows Oerson shot over the bar and Sandile Ndlovu had his shot blocked.
Two late substitutions by Ezenkosi boss Jomo Sono failed to produce the desired result as he had to settle for a point, leaving his side still entrenched in the relegation zone, five points drift of 14th placed Thanda Royal Zulu.