Spieth and Huh both caught Reed at 14 under down the stretch in regulation.
At roughly the same time Reed bogeyed the par-3 16th, Spieth birdied the par-4 17th and closed with a par. Huh quickly slipped off the pace after finishing with two bogeys.
Reed, who led or shared the lead after the second and third rounds, also could have won it in regulation after landing his approach shot on the 18th in the center of the green, but left his approximately 20-foot birdie putt short.
For the second straight day, organizers tried to beat the rain by starting the round early, sending players off in threesomes from the first and 10th tees. The skies were ominously overcast all day, but the saturated course didn't receive any rain.
And unlike the third round — in which only 13 players broke par — scores were significantly lower on the water-logged Sedgefield course, and that turned the final 18 holes into a shootout.
Jones birdied five consecutive holes and six of seven during his best round of the year.
"Every golfer out here can go and shoot that," Jones said.
Jones and Simpson, the 2011 winner who shot a 63, led the 52 players who shot better than even-par 70 during the final round.
"You really couldn't tell that the greens got any rain," Simpson said. "They were still as fast today as I've ever seen them."
It wound up being a mostly fruitless week for the players on the playoff bubble who missed their last chance to push their way into The Barclays next week.
Nobody who started the week outside the top 125 managed to make it in. Each of the players at Nos. 126-132 missed the cut, and No. 133 Robert Streb finished at 3 under but could only jump to 126th.
"I was trying to put (the pressure) to the side as much as I could," Streb said. "You can't completely ignore it, but I just tried to play the best golf I could."
Follow Joedy McCreary on Twitter at http://www.twitter.com/joedyap
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