The 23-year-old Reed fired a final round of 66 while Spieth carded a five-under 65 to leave the two players tied at the top of the leaderboard on 14 under at Sedgefield CC.
A play-off followed but the players could not be separated on the first hole either, with both hitting pars on the 18th.
The pair then moved to the 10th and a second play-off hole, and although Reed drove into the trees, he found the green with his second before sinking a birdie which Spieth was unable to match.
Reed's final round saw the Texan card five birdies and a single bogey, while 20-year-old Spieth overcame a dropped shot on the opening hole by sinking six birdies.
That left the players two shots clear of the rest of the field, with Brian Harman and John Huh the nearest challengers on 12 under.
Harman also ended the tournament with a bogey-free round of 66, while Huh mixed five birdies with three bogeys to sign for a two-under 68.
Matt Jones was the highest placed non-American, the Australian in a three-way tie for fifth after a superb final round of eight-under 62 to finish alongside Matt Every and Zach Johnson on 11 under.
Reed was unable to hide his surprise at his astonishing comeback in the play-off, having been so close to going out of bounds with his tee shot before somehow landing his ball seven feet from the hole with his second.
"It was the best shot of my life, that's for sure," Reed said on www.pgatour.com.
"Just to get my first win means everything to me."
There was no last-minute breakthrough from any players either in a bid to reach the FedEx Cup play-offs.
The top 125 on the points list after this weekend's event all qualify for The Barclays next week, but late pushes from Jim Herman (nine under) and Ricky Barnes (six under) fell short with the pair finishing 139th and 132nd respectively.
Robert Streb came the closest to making the cut, finishing the tournament on three under to move up from 133 to 126 on the list and one place short.