City made one change from their win over Royal AM in midweek as Khanyisa Mayo was preferred ahead of Douglas Maphumo, while Swallows named Strydom Wambi as their only change too.
An enterprising opening found chances either end earlier on, with Terrence Mashego firing a smart opening across goal before Junaid Sait came close with a close-range header at the other end.
Wambi was presented a clear chance to make the breakthrough before the 20th minute spell from a low pass by Khetukuthula Ndlovu although after twisting about, shot tamely at Hugo Marques.
The Citizens launched a clearer threat where Abbubaker Mobara ran in at the back-post six minutes towards the half-hour mark to side-foot Mayo's pin-point free-kick against the woodwork.
Swallows should've trailed in stoppage time from a powerful left-footed shot by Mayo, but Virgil Vries tipped for a corner-kick before Taariq Fielies headed the resultant corner wide.
A competitive start to the second half saw a block from Mobara deny Dillon Solomons a route to goal, while a similar parry by Sait and Sipho Sibiya on the other side smothered Mduduzi Mdantsane.
City continued to enjoy the lion share of chances as a flowing one-touch move near the latter stages by Mashego coincided with Mayo, who brought a low save out of Vries with a first-time shot.
The Dube Birds rolled out two substitutions with under 20 minutes to go in Mwape Musonda for Wambi and Mphakamiseni Nene introduced ahead of Lehlogonolo Matlou.
Eric Tinkler's side enjoyed a strong finish with Vries forced to parry a low shot by Abdul Jeleel Ajagun, while a move by Keletso Makgalwa had City on alert but goalless it finished.
City: Hugo, Mkhize, Mobara, Fielies, Mashego, Makola, Ajagun, Mdantsane, Ralani, Martin, Mayo.
Subs: Anderson, Fasika, Mokeke, Dove, Mapfumo, Goedeman, Mokhuoane, Roberts.
Swallows: Vries, Matlaba, Sait, Nyatama, Sasman, Sibiya, Matlou, Ndlovu, Wambi, Solomons, Gamildien.
Subs: February, Musonda, Nene, Mohamme, Mbokoma, Khupe, Makgalwa, Basadien, Mere.