Salman, who earlier had career-best bowling figures of four for 32 as South Africa were restricted to 239 for nine, was named man of the match – but he handed the trophy to Ayub.
“Without Ayub, we would not have won the game,” said Salman. “He was there with the new ball and set up the game.”
There were still moments of anxiety for the tourists after Kagiso Rabada had Ayub caught at fine leg with 38 runs still needed.
Two more wickets fell before Salman hit the winning boundary with three balls to spare.
Heinrich Klaasen made 86 off 97 balls in South Africa’s total of 239 for nine but was forced to play more conservatively than usual after the innings stalled against Salman’s off-spin.
South Africa made a rapid start with Tony de Zorzi (33) and Ryan Rickelton (36) putting on 70 for the first wicket inside the 10-over power play.
But Salman, who had taken only 10 wickets at a cost of 50 runs each in 27 previous one-day internationals, took the first four wickets in the space of 12 balls as South Africa slumped to 88 for four.
Spin bowling dominated, with slow bowlers sending down a combined 27 overs and taking seven wickets for 107 runs.
Ayub, 22, followed up an unbeaten 98 in a losing cause in a Twenty20 international in Centurion last Friday with an innings of skill and maturity,
Ayub and Salman stabilised the innings before steadily increasing their scoring rate.
The match swung dramatically in Pakistan’s favour when Ottneil Baartman returned to the bowling attack in the 35th over. Baartman had taken two for nine in five overs in his first spell but Ayub hooked his first two balls for six in an over which yielded 22 runs.
South Africa suffered a blow during the warm-up when Keshav Maharaj, their premier spin bowler, suffered a groin strain and had to withdraw.