"It was a tricky chase," said Bavuma. "They got the ball to talk with the new ball, it was tricky. I was just trying to hold in and allowed Klassen to come and do what he does best."
Kumar, who returned figures of 4-13, took down Reeza Hendricks and Dwaine Pretorius in his first two overs and then bowled previous match hero Rassie van der Dussen, who made 75 in his team's opening win.
Klaasen soaked up the pressure and the humidity with his clean hitting.
He raised his 50 in 41 balls and took apart the Indian attack to make the target look simple with the in-form David Miller for company.
Klaasen hit seven fours and five sixes before departing, but the job had been done and the left-handed Miller hit the winning runs to finish with 20 not out.
Rabada's 50th T20I scalp
Klaasen came in ahead of Miller, who played a key part with his 64 in the opener, at number five but Bavuma said his batsmen justified their positions.
"David, we can use him at five or six. Him obviously being a left-hander, it helps in the end. Klaasen is someone who can with a couple of balls do a lot of damage," said Bavuma.
"Klaasen has showed there he adds a lot of value to our batting. It's about whatever you role is, whatever the situation is, execute as well as you can."
Earlier Rabada sent back Ruturaj Gaikwad for one to bring up his 50th T20 wicket for South Africa and returned figures of 1-15.
Left-handed opener Ishan Kishan (34) and Shreyas Iyer (40) then attempted to rebuild the innings in a 45-run partnership that was broken by fast bowler Anrich Nortje, who took two wickets.
Left-arm spinner Keshav Maharaj sent back skipper Rishabh Pant in his first over and soon Hardik Pandya's wicket off Wayne Parnell made India slip further.
But Dinesh Karthik hit an unbeaten 30 off 21 balls to give the team a late push during his 36-run unbeaten stand with Harshal Patel but the effort was not enough.
The third match is in Visakhapatnam on Tuesday.