Ramsay shot a 7-under 65 on Sunday to move past overnight leader Pablo Martin of Spain, who shot a 73 at Jack Nicklaus-designed Pearl Valley. Sunshine Tour Order of Merit winner Anders Hansen of Denmark was a stroke back in third after a 69.
Kapur finished with a 67 in the event sanctioned by the European and Sunshine tours.
Ramsay had a putt on No. 18 to win the championship, but it went just left, forcing the playoff.
Kapur had to lay up in the playoff, also on No. 18, while Ramsay reached the green in two and his 15-foot eagle putt came up just short, enabling him to tap in.
"I had no thought of laying up, because I was so sure of myself over the shot," Ramsay said.
The Scot took control at the 17th went he sent his approach over a water hazard and landed it on the green to leave a 15-foot putt.
"I hit an 8-iron in, and I couldn't believe it cleared the water," he said. "My heart was in my mouth."
Martin, who won last week's Alfred Dunhill Championship, bogeyed the third and never recovered, managing only one birdie. A bogey at the 17th saw him finish tied for sixth with Darren Fichardt of South Africa (66).