After the 251st-ranked Storm tapped in for a par, McIlroy slid his par putt wide from 7 feet on their fourth visit to the 18th hole at the Glendower Golf Club.
''I'm in shock, this has been a surreal week,'' Storm said. ''To find myself in the position I was in, playing on the final day with the best player in the world right now. It's just a dream come true.''
McIlroy, the world No. 2, started the final round three strokes behind Storm but chased down the overnight leader, moving atop the leaderboard when Storm missed a 3-foot par putt on No. 14. McIlroy relinquished the lead by bogeying No. 17 after taking two shots in a greenside bunker, sending the event to a playoff with both at 18-under 270.
McIlroy shot 4-under 68 and Storm had a 71.
Storm lost his card at the end of last year, only to get a reprieve when American Patrick Reed failed to play enough events to join the tour.
His other title came at the French Open in 2007.
Tour rookie Jordan Smith of England was a shot back in third, ahead of a trio of South African players. Dean Burmester was fourth on 273, one stroke ahead of Thomas Aiken and Trevor Fisher Jr.
On the first playoff hole, Storm sank a close-range putt for par. They went back up the par-4 18th and both players drove into the rough but still managed to make par.
The third time round, McIlroy hit his approach shot short of the green to give Storm the advantage. The Englishman's 45-foot birdie putt just missed, as did McIlory's putt for par minutes later.
Storm played cautiously Sunday, coming up short with many putts on the back nine to allow McIlroy to eat into his lead.