Shaheen picked up 3-26, being the standout of Pakistan’s bowling attack in which Faheem Ashraf and debutant Usman Tariq picked up two wickets each, as South Africa were restricted to 139-9 after being put into bat.
Babar then provided anchor in the chase with a brilliant 68, featuring nine fours, as Pakistan eased to victory with six balls to spare.
Shaheen got Pakistan off to a flyer with two wickets halfway through the first over and almost bagged a third victim two balls later.
Opener Quinton de Kock’s stumps were undone on the second ball of the innings with a delivery that jagged back, found an edge and clattered into the woodwork.
On the very next ball, Lhuan-dre Pretorious then spooned a catch to Usman Tariq at short fine-leg.
Pakistan were in dreamland when the umpire adjudged Dewald Brevis lbw on the first ball of a spectacular opening over only for the decision review system to show that the ball was going to rise over above the stumps.
South Africa were rocking and their first boundary only came on the penultimate ball of the third over when opener Reeza Hendricks punched Shaheen away to the extra cover fence.
Things were otherwise kept tight by Shaheen’s new ball partner Salman Mirza, coming off a sparkling performance in the second match of the series a day earlier where Pakistan had shot down South Africa for 110 to level the series, giving little away.
Faheem, player-of-the-match on Friday, gave away just two runs in the fifth over with Salman bowling a tidy sixth as South Africa crawled to 22-2 at the end of the powerplay.
Brevis, though, broke the shackles with two sixes off spinner Mohammad Nazwaz in the seventh over — the first straight down the ground and the second over deep midwicket.
Brevis (21), though, was gone soon after with debutant spinner Usman Tariq getting his maiden T20I wicket with the second ball of his career; the South African batter finding Babar Azam at long-on as he tried to go big once again.
The incoming Matthew Breetzke didn’t last long, seeing his stumps shattered by Nawaz with a ball that straightened after pitching in the following over as South Africa slipped from 38-2 to 42-4.
Skipper Donovan Ferreira breathed fresh impetus with a six off Usman, as South Africa reached 56-4 halfway through their innings, and then upped the ante with two sixes and a four in the 11th over bowled by Nawaz.
But Ferreira’s big-hitting spree didn’t last long as, after making 29 off 13, he holed out to his counterpart Salman Ali Agha at mid-off in the following over by Faheem.
South Africa were six down on the next ball with George Linde finding Hasan Nawaz at midwicket with Pakistan firmly on top.
Hendricks’ long, patient stay at the crease ended in the 15th over when he top-edged Usman to Hasan at deep backward square to depart for 34 off 36 with two fours.
The incoming Andile Simelane then hoisted Shaheen for the fourth six of the innings to set South Africa’s total beyond Friday’s below-par effort but departed after making 13 when Salman Mirza had him caught at midwicket in the 18th over.
Shaheen bagged his third wicket when he sent back Lizaad Williams in the penultimate over but Corbin Bosch (30 not out) ensured South Africa reached a respectable total.
Sahibzada Farhan got Pakistan’s first runs in the chase with a six over long-off on the third ball of their innings but lost his opening partner Saim Ayub in the second over — the left hander miscuing to Ferreira of Bosch.
Babar, arriving to adoring cheers from the crowd, got his first four on the ninth ball he faced with a sumptuous pull off Williams and then cut him away for another two deliveries later.
Babar was in the mood and swept Linde for a one-bounce four in the following over as Pakistan reached the end of the powerplay at 36-1.
Sahibzada then got his second boundary, driving Ferreira down the ground for four, but departed soon after when the South African skipper had him caught at deep midwicket for 19 on the last ball of the seventh over.
In the face of spin from both ends, Babar and captain Salman Ali Agha sought to steady the chase with just one boundary coming in overs eighth to 10th, by the end of which Pakistan were 64-2.
Boundaries, however, came thick and fast in the 12th over by Ottneil Baartman — Agha’s edge flying to the third man fence after a misfield before Babar got three in a row to raise his half-century off 36 balls.
Babar, who returned to the side for the series, was putting on a show and got another two boundaries in the 14th over of Bosch as Pakistan closed in on the target.
Babar and Salman, who was providing able support at the other end, however couldn’t take Pakistan to the end.
Salman had just got his second boundary but perished on the next ball, having made 33 off 26, when he chipped Williams to Simelane in the 16th over.
Babar fell in the next over, when he pulled Bosch to Hendricks at deep backward square to bring an end to his 47-ball knock.
Babar’s fall was followed with two more wickets as Hasan was caught off Williams and Silemane cleaned up Nawaz but Pakistan eventually wrapped it up with an over to spare.