Streb sunk a four-foot birdie putt at the second play-off hole to edge fellow American Will MacKenzie and Zimbabwean Brendon de Jonge at the Seaside Course in Georgia.
De Jonge missed his 35-foot birdie attempt at the par-three 17th beforehand, while MacKenzie was eliminated at the first additional hole after only managing a bogey. Streb's triumph sealed his place at next year's Masters.
The trio finished level at 14 under at the end of the 72 regulation holes.
Streb birdied four of his final five holes as he carded a seven-under-par 63 to surge 11 positions up the leaderboard, while overnight leader MacKenzie and De Jonge posted rounds of 68 and 65 respectively.
American quartet Kevin Kisner, Ken Duke, Chris Kirk and Russell Henley tied for fourth at 12 under.
Kisner produced a bogey-free five-under-par 65 and Duke shot a final-round 66.
Kirk - the Deutsche Bank Championship winner last month - had a 67, while Henley could only manage a 69 on the final day.
A shot further back were Kevin Chappell (65), Mark Wilson (67), Fabian Gomez (68) and Andrew Svoboda (71).
Svoboda was co-leader overnight but faltered dramatically in the final round, bogeying three of his remaining eight holes.
Hudson Swafford (67) and Andrew Putnam (69) ended the tournament in a tie for 12th at 10 under.