England looked out of the game for much of the chase with Jos Buttler, Phil Salt, Moeen Ali and Jonny Bairstow all caught cheaply to leave them 61-4.
By the end they could easily have won it.
The pendulum swung England’s way when Ottneil Baartman bowled five full tosses in the 17th over which cost 21.
That left England, who earlier needed 104 from the last 10 overs and 77 from the last six, needing 25 from 18 balls.
But Liam Livingstone hit another full toss, this time from Kagiso Rabada, to deep square-leg, in the next over, with the South Africa quick only conceding four singles.
Marco Jansen did not concede a boundary in the penultimate over but still, with Brook at the crease, England were right in the game.
It was fitting the decisive moment was a piece of superb fielding, in a match packed with such efforts.
Brook tried to clear long-off but captain Markram ran back, leapt and held onto the catch.
Curran turned down a single in the dying moments, backing himself to clear the ropes, but Nortje executed his plans to close out the win.