Their first victory means they are lying in third spot ahead of the final games, a point adrift of Japan and Sweden, whom they meet on Tuesday at St James' Park. Canada, beaten 2-1 by Japan in their opening match, went ahead after only seven minutes.
Melissa Tancredi, who had netted against Japan, was again their marksman when converting a low centre from Diana Matheson. Tancredi was a constant threat and headed just wide at the near post from a Lauren Sesselmann cross, while Kaylyn Kyle had a powerful effort saved by Thokozile Mndaweni.
South Africa came more into the game before half-time, with Portia Modise's shot on the turn saved by Karina LeBlanc. Then hesitation by Robyn Gayle let in Mpumi Nyandeni only for the Canadian defender to recover and deflect the shot onto the crossbar.
But Sinclair scored her 138th goal for Canada after 58 minutes. Her header from Sesselman's centre hit the underside of the bar but had clearly crossed the line. Canada threatened to add to their lead and a 25-yard drive from Matheson struck the crossbar.
With four minutes left, Sinclair broke clear and slid the ball past Mndaweni for her second goal and Canada's third.