Weekley rolled in birdies on Nos. 8 and 9 and followed those up with two more on holes 10 and 13, finishing 72 holes with a one-stroke lead at 14-under-par 266.
It was the third career PGA Tour victory for the 39-year-old Florida native.
Second- and third-round leader Matt Kuchar fell off the pace late but managed a birdie on the final hole to end one shot back at minus-13 with a round of 2-under 68.
Zach Johnson (66) claimed sole possession of third place at 12-under 268 at Colonial Country Club.
Sharing fourth place were Scott Stallings (66), John Rollins (68) and Matt Every (69).