The hosts posted a score of 146-7, with Aussie debutant Pat Cummins doing most of the damage with the ball.
The teenager claimed three wickets in the 19th over to take the South Africans from 133-3 to their final total.
The visitors' innings began disastrously when David Warner went for a duck off the third ball.
Watson steadied the ship with a knock of 52, and scores from Shaun Marsh , David Hussey  and Cameron White  pushed Australia closer to their target, which Steven Smith reached with three balls to spare.
The second match of the two-game series takes place on Sunday in Johannesburg.