The 44-year-old, who was born in Czechoslovakia and moved to Germany when he was nine, qualified for the play-off by carding 69 in the final round of the Puerto Rico Open.
The score, which included birdies on four of the first six holes, put Cejka on seven under par for the tournament and level with Sam Saunders, Jon Curran, Tim Petrovic and Emiliano Grillo.
Cejka, who shot to victory clad in black and white golf clothing , becomes the first non-American to win the tournament and the first golfer born in the Czech Republic to win a PGA Tour event.
The five players taking part in the play-off had finished a shot ahead of US golfers Boo Weekley, Scott Pinckney, Will MacKenzie and Will Wilcox.
Scott Brown and Chris Smith had gone into the final round leading the tournament but both had their challenges derailed by a series of bogeys.