Both players finished 72 holes at 18-under-par 270.
Fasth, who closed with a 3-under 69, collapsed in the late going. After building a two-shot lead through 16 holes, he bogeyed 17 and 18. Bickerton was in the clubhouse after a course record-tying 63.
"It was a bit more work than I was hoping for," said Fasth, who also won playoffs for his two tiles in 2005. "I really had it in my own hands and messed it up a bit."
Thomas Bjorn (69) of Denmark ended in six-way tie for third place at 17-under-par 271 with Sweden's Mattias Eliasson (67), David Griffiths (72) and Phillip Archer (67) both from England, Jose Manuel Lara (68) of Spain and Gary Orr (67) of Scotland.
Fasth won $275,000.