In an improbable finish, Byrd came from four shots down with five holes to play. He made three birdies in a four-hole stretch from 14-17, to shoot a closing 5-under 66.
His final total was 18-under 266. It was Byrd's third PGA Tour victory and his first since the now-defunct B.C. Open in 2004.
"I'm going" Byrd said about next week's British Open. "I don't have any other plans. I just have to make a quick trip home to get my passport."
The victory was worth $738,000.
It was a difficult day for Clark, who closed with a 3-under 68 for a final total of 17-under 267. He was the runner-up at the 2006 Masters.