Coming to the crease at 27-2, chasing 180 to win, Bairstow hit nine fours and four sixes in his match winning knock.
He was supported in a 85-run partnership with all-rounder Ben Stokes, who made a 27-ball 37.
Earlier recorded his best T20 international bowling figures as he took 3-28, including the key wicket of Faf du Plessis who top-scored for the hosts with 58.
South Africa’s total of 179-6 was 21 more than the average first-innings T20 total at Newlands, and England were struggling at 34-3, but Bairstow’s brutality guided them to an excellent win with four balls to spare.
The second game of the three-match series takes place at Boland Park, Paarl on Sunday at 12.30pm GMT.
England return to Edgbaston in 2021 as the world champions take on Pakistan in a day/night ODI on Tuesday 13 July (provisional date).
Enter the Ballot: England v Pakistan ODI
Enter the ballot before 5pm on Friday 11 December for a chance to secure tickets for the ODI between England and Pakistan at Edgbaston on Tuesday 13 July.
Demand is expected to be high, so enter the ballot today.