Perry, who held at least a share of the lead after each of the three rounds, closed the final round with a 5-under 67, as did Langer. Each finished 54 holes tied at 13-under-par 203.
On the extra hole, Langer's approach shot missed the green right and he followed with a chip shot to within 8 feet of the pin.
Perry placed his second shot 9 feet away and then drained a birdie putt for his third title of the season.
Second-round co-leader Colin Montgomerie started quickly with birdies on five of first six holes, but then faltered, finishing with a 3-under 69 to end up in third-place tie at 11-under with Kirk Triplett (68) and Fred Funk (67).