In the final 18 holes of regulation, the 2006 U.S. Amateur champion shot 7-under-par 65, while Kapur shot 5-under 67, with both ending 72 holes with totals of 13-under 275.
The par-5 18th was the playoff hole, where Ramsay, a native of Aberdeen, Scotland, two-putted for birdie and his first European PGA Tour title.
"There's no greater feeling than winning," Ramsay said. "I'm just a flood of emotions. … I'm holding back the tears. (Now) I go out believing and thinking I can win."
Kapur, from New Delhi, qualified for the playoff with five straight pars at the end of regulation.
Spaniard Pablo Martin (73), the leader after the second and third rounds, finished 10-under 278 and shared sixth place with England's Darren Fichardt.