Bourdy, Alastair Forsyth and David Howell were tied at 18-under 266 after four rounds at the Oitavos Dunes course, with Bourdy carding a final-round 7-under par 64 to pull into a tie.
All three parred the first extra hole, No. 18, sending the trip back to the same hole. Forsyth dropped out with a bogey, and Bourdy and Howell then went to No. 17.
Bourdys approach shot landed about 30 feet from the hole, while Howell left his shot left of the green. Howells chip left him with a 20-foot par putt, while Bourdy rolled his third shot with 18 inches.
Howells par put slipped past the hole, and the 25-yerar-old Frenchman tapped in for the win.
It was Bourdys second European Tour title.
Miles Tunnicliff of England finished fourth at 267, while Spains Gonzalo Fernandez-Castano finished fifth at 268.