Weekley carded a second straight four-under-par 66 to finish 14 under overall, one clear of Kuchar, who could only manage a 68 and finished second having led after the second and third rounds.
"Finally made it back," Weekley, whose two previous titles were at the RBC Heritage in 2007 and 2008, told pgatour.com.
Weekley was rewarded for his consistency having shot rounds of 67, 67, 66 and 66 this week.
Four adrift at the halfway stage of the tournament, he crept up on Kuchar and got the job done early in his final round.
After birdies at one and three, he bogeyed the fifth and seventh, but picked up further shots at eight, nine, 10 and 13 to establish a lead which proved decisive.
He missed out on decent birdie chances at the 14th and 17th, but with nearest challenger Kuchar struggling to make any impression he had a little breathing space.
Kuchar was level through 11 before picking up shots at the at the 12th and 18th, but it was too little too late to challenge Weekley.
"I finally get my name on that wall with Ben Hogan and some of the greats that have played here," Weekley told the Golf Channel. "It's awesome."
Last year's champion Zach Johnson carded birdies at the 16th and 17th to complete a 66 of his own and finished third on 12 under.
"I played solid," Johnson told pgatour.com. "What I did today was I gave myself a number of opportunities. It seems like almost every other hole I had a really good birdie look. I left some out there, but I also made some putts, so it probably all evens out."
South Africa's Tim Clark finished in a share of seventh on 10 under to gatecrash an otherwise all-American top 10.
Sweden's Jonas Blixt was the leading European on nine under, while England's Brian Davis slipped back to two under after a final-round 73, two strokes ahead of Scotland's Martin Laird.