The title was the fifth on the PGA Tour for the native of South Africa.
Sabbatini birdied the 11th hole to take the lead for good, then notched four more birdies coming in.
Brian Davis, also with a 64, placed second at 263.
"It is a good feeling obviously winning anything associated with Lord Byron Nelson," Sabbatini said.
Hall of Famer Nelson died in 2006 at the age of 94.
Sabbatini collected $1.17 million.