The speedy winger handed AmaTuks a 2-0 lead at the break with two stunning efforts from outside the area, before Thabo Mnyamane made sure of all three points with five minutes remaining.
The hosts ended the game with 10-men; club captain Tebogo Monyai seeing red following a second bookable offence in the 88th minute.
In an attempt to combat the attacking powers of Rise and Shine, Tuks head coach Sammy Troughton named three recognised strikers in his starting XI, with Thabiso Nkoana and Getaneh Kebede making their home debuts, while Atusaye Nyondo returned to the line-up from injury.
Ronald Ketjijere and Vuyisile Ntombayithethi were tasked with keeping the likes of Puleng Tlolane quiet.
Tuks’ attacking intent was visible from the opening minute, with a succession of corner-kicks going unrewarded.
Tlolane thought he had a shout for a free-kick and a possible sending off within six minutes after he took a tumble just outside the area, but the referee instead brandished the yellow card for diving.
Kebede saw his backheel fly just wide in the 17th minute, but the home supporters would be on their feet soon after. City looked to have stifled a Tuks attack, but Kebede rolled the ball into the path of the onrushing Zwane, who opened the scoring with an unstoppable shot in the 31st minute.
Zwane doubled his tally seven minutes later with a rasping effort from the edge of the area which left Pule Konopi with no chance.
Polokwane's best chance of the game would fall to Linda Shiba just after the break; the second-half substitute hitting the post from an acute angle.
Mnyamane made sure of all three points in the 85th minute - the lanky forward with a neat finish within a minute from coming off the bench on his first game of the season.