Pakistan elected to bowl first and restricted the visitors to 139 for nine in their 20 overs, before chasing down their victory total with six deliveries remaining.
Babar had looked out of touch in the first two games in the series on his return to the Pakistan T20 side for the first time this year, but conjured up a match-winning effort on a tricky surface that was providing assistance to the seamers.
He put on 76 for the third wicket with captain Salman Agha (33 from 26 balls) that set up the victory for the home side.
"This innings was due," Babar said at the post-match presentation. "I backed myself and the team believed in me. I was hoping for such a knock.
"There is pressure in everything. It is about how you absorb it and I wanted to do what the team needed, play according to the situation.
"We wanted to take it deep and build partnerships and it worked for the team."
Several South African batters made starts in their innings but wickets fell at regular intervals.
Reeza Hendricks top-scored with 34 and Corbin Bosch (30 not out) and captain Donovan Ferreira (29) were the others with modest contributions.
Pakistan seamer Shaheen Shah Afridi did early damage with two wickets in the opening over to put the visitors on the back foot and finished with figures of 3-26.
The two teams begin a three-match One-Day International series in Faisalabad on Tuesday.
"We didn't perform with the bat once again," Ferreira said. "The bowlers did well to fight until the end but there were not enough runs on the board."