Saim Ayub returned to form with a belligerent unbeaten 71 off 38 balls as Pakistan raced to 1-112 in just 13.1 overs after bowling out South Africa for 110. Fast bowlers Salman Mirza (3-14) and Faheem Ashraf (4-23) shared seven wickets.
The third and deciding T20 is on Saturday.
"They outplayed us in the previous game and we did that today," Pakistan captain Salman Ali Agha said.
"It's beautifully set up now and hopefully we'll have a cracker in the final. We bowled really well up front and when you bowl that well you're always going to win the game."
Babar needed nine runs to pass Sharma's record of 4231 and finished 11 not out. He overtook the Indian batter's record when he drove spinner Donovan Ferreira for a single to long off.
Pakistan recalled Babar for his first T20s in almost a year after Fakhar Zaman was rested. Babar missed the chance in the first match to set the record when he was dismissed for a two-ball duck in Rawalpindi on Tuesday.
Babar has 4234 runs in 130 T20s, including 36 half-centuries and three hundreds, but his strike-rate of 129 has often been criticised.
He missed the Asia Cup in which Pakistan were beaten by India.
Sharma played 159 T20s but quit international cricket's shortest format after leading India to the T20 World Cup title last year.
MOST RUNS IN MEN'S T20s:
* 4234 - Babar Azam (Pakistan, 2016-25)
* 4231 - Rohit Sharma (India, 2007-24)
* 4188 - Virat Kohli (India, 2010-24)
* 3869 - Jos Buttler (England, 2011-25)
* 3710 - Paul Stirling (Ireland, 2009-25)
* 3531 - Martin Guptill (New Zealand, 2009-22).
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