Verplank, a native of nearby Dallas, finish at 13-under 267.
Donald (68), the overnight leader, was alone in second place at 12-under 268.
"I've never been so nervous in my life on that last little putt," said Verplank, who claimed a winner's check of $1.1 million for his fifth PGA Tour victory. "Byron meant so much to me and to be from Dallas and winning this -- it's unbelievable."
Nelson died last year.
"I know he'd be so proud," said Peggy Nelson, Byron Nelson's widow. "Byron would be very happy for Scott. It's been such a long friendship that they've had and it's nice to see it culminate this way."