The teams will play three T20s and three one-day internationals on England's short-notice tour, organised to give both teams a chance to play some cricket in a virus-disrupted worldwide schedule.
Bairstow fired from his new No 4 berth to rally the tourists from 34-3 and lead them to their victory target of 180 with four balls remaining, with his fourth six - allied to nine fours - sealing his team's win.
Today marks the first match of England's long and winding worldwide winter that will take them from South Africa to Sri Lanka and India, beginning with three T20s as they tune up for next autumn's World T20 in India.
Ben Stokes looks certain to play as one all-rounder, but whether England will play another in the number seven batting position is unclear.
Former captain Faf du Plessis top-scored for South Africa with 58 off 40 balls after his side were sent in on a cool evening.
Sam Curran took 3-28 for England and helped out with the bat with a six of his own right at the end, and was 7 not out.
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The left-armed Curran struck in the first over of the match when Temba Bavuma gloved a catch to wicketkeeper Jos Buttler, attempting a lap shot.
"The partnership will kick off with the England tour of South Africa on 27 November 2020 during which the three T20Is will be broadcast in Hindi for the very first time". South Africa A team coach, Malibongwe Maketa has also joined the Proteas camp as extra cover to assist the team.
With leg-spinner Adil Rashid a consistent threat in white-ball cricket, England may have a slight advantage in spin bowling in the absence of iconic spinner Imran Tahir, who is no longer in the hosts' squad. Even so, South Africa still had a realistic chance of winning as England went into the last four overs needing 51 runs.
However, Hendricks' wide and boundary-infested 17th over changed that to 23 from 18 and although Eoin Morgan (12) holed out to deep midwicket off Ngidi in the 18th, Sam Curran nailed a six off Kagiso Rabada in the 19th and Bairstow sealed matters in the 20th.
As was confirmed during the week, neither side was expected to take a knee in support of the Black Lives Matter movement before the game began, though two sizeable banners were visible in the stands.