South Africa had the better of most of the first three days but began the fourth day 27-3 with their hopes in the balance.
Experienced pair Bavuma and Aiden Markram settled early nerves until Abbas bowled Markram for 37 for his first wicket of the fourth day.
Abbas, with his immaculate lines and lengths, bowled throughout the 30.3 overs of play either side of lunch on a lively pitch.
The Proteas’ innings hit real strife after Bavuma was given caught behind off the seamer for 40, although replays suggested the ball flicked his pocket rather than bat, potentially saving the South Africa skipper had he chosen to review.
Naseem Shah then bowled Kyle Verreynne for two and Abbas had David Bedingham and Corbin Bosch caught behind in the next over.
At that stage South Africa needed a further 49 with two wickets left as Rabada joined all-rounder Jansen.
First the pair reached lunch at 116-8, calming some of the drama. Afterwards Rabada played the decisive hand.
The left-hander attacked and crashed four of his five fours after the break.
It swung the momentum back to South Africa and when Jansen steered the winning runs behind point there were scenes of jubilation in the stands and obvious emotion among the players.
“There wasn’t much conversation,” said Bavuma, asked about the situation at lunch. “I was still sulking in the toilet. I came out when we needed about 15 runs. It was a tough one.”
