Polokwane, though, could still be sucked into the promotion/relegation playoff spot currently occupied by Free State Stars, who are two points behind.
The teams struggled to get going in an attacking sense and the first notable effort at goal only arrived midway through the first half, with Thapelo Tshilo heading wide from a corner for the home team.
Polokwane had plenty of dead-ball situations with which to continue threatening the Birds’ defence, but they were unable to provide the quality of delivery to give their attacking players scoring opportunities.
City made the first change 12 minutes into the second stanza, with Thabo Rakhale replacing Thobani Mncwango in an attempt to liven up their wing play.
On 65 minutes Swallows created their first chance of the match, with defender Luvhengo Mungomeni shooting wide of the target from a Left Tsutsulupa free-kick.
That was to be the last notable opportunity of the match, which fizzled out.
Polokwane will finish the season away to SuperSport United on Saturday, while Swallows are home to Free State Stars.