After Goydos found the water with his tee shot at the first extra hole, Garcia punched a wedge to the middle of the green and watched as his ball fed down to the hole, nestling only four feet away.
And after Goydos missed his unlikely par attempt from the drop zone, Garcia could celebrate the end to a drought of nearly three years after tapping in for a par.
Earlier, Garcia made a par on the final regulation hole to stay alive.
The par-four 18th was the toughest hole on the course on an extremely windy day, and Garcia found a poor lie when his tee shot drifted into the rough.
He advanced his second shot to within 50 yards and then pitched to seven feet before holing the pressure putt to card a final-round 71.
Goydos (74), playing in the group behind, missed his chance to win in regulation when his putt drifted to the right. Both players were tied at five under par.
American Jeff Quinney (70) bogeyed the last to miss the play-off after his approach shot found the back bunker.