Chasing a modest 119 for victory, Australia were in trouble at 81-5 after Steve Smith, who scored 35, and Glenn Maxwell, who made 18, departed in Abu Dhabi.
Stoinis, who made 24, stood firm and put on an unbeaten stand of 40 with the left-handed Wade, who made 15. Stoinis hit the winning boundary to take Australia over the line with two balls to spare.
Anrich Nortje took two wickets including skipper Aaron Finch, for nought, and Smith to return figures of 2-21 and make Australia work for their runs.
Tabraiz Shamsi dismissed Maxwell as the batsman’s attempted reverse hit ended with the ball crashing into the stumps.
Earlier, pace bowler Josh Hazlewood took 2-19 to help limit South Africa to 118-9 after the Aussies elected to bowl first.
Number four Aiden Markram played a lone hand with his 40 off 36 deliveries before falling to Mitchell Starc.
Markram kept losing partners with Heinrich Klaasen and David Miller departing for 13 and 16.
Leg spinner Adam Zampa struck twice in one over and a big mid-pitch mix up led to Keshav Maharaj being run out for nought.
Kagiso Rabada hit a an unbeaten cameo of 20 off 23 balls to take the Proteas past 100 .