The 27-year-old Reed battled to a one-under-par 71 for 15-under 273 to win by a stroke from Fowler, who birdied the last in a dramatic finish for a 67.
That left Reed needing to par the final hole, draining a testing putt to close out his first major victory before being embraced by his wife Justine.
"It's almost impossible to put into words how I feel," said Reed, who completed his college golf career at nearby Augusta State University.
Spieth finished a further shot back after closing with a stunning eight-under 64. He had started the final round nine shots behind Reed, but quickly cut into the deficit with five birdies in his outward half.
Reed still had a comfortable buffer but 2015 winner Spieth stepped up his challenge with a further run of four birdies in five holes.
The start of his back nine charge came at the short 12th where two years ago he imploded with a quadruple bogey to hand victory to England's Danny Willett.
Finding the back of the green, Spieth raised his club ironic celebration before draining the putt and starting his birdie blitz, culminating in a long-range effort on the 16th to get to 14-under.
But a final hole bogey after a dreadful tee shot blunted his challenge and left Fowler to take up the mantle with four birdies in his last six holes, coming up just short in his bid for an elusive first major.
Spain's Jon Rahm briefly threatened on the back nine, but finished four shots behind in fourth.
The expected challenge from Rory McIlroy never materialized, his hopes fading after he missed an eagle putt on the second which would briefly have drawn him level with Reed at the top of the leaderboard.
Other similar misses followed, leaving the Northern Irishman to contemplate another disappointing failure at Augusta as he finished tied for fifth on nine-under after a 74. He will have to wait another year at least to complete his career grand slam.
Woods finishes strongly
The early part of proceedings were enlivened by Tiger Woods' best round of the tournament, a three-under-par 69 for one-over 289.
Woods, who was never able to move into contention after an opening 73, birdied the 13th and then eagled the 15th to a huge roar from the galleries.
It left him in a tie for 32nd but showing promise he could be a factor in the remaining three majors as he bids to improve on the 14 he has already won, but the last in 2008.
Woods is just grateful to be given the chance to compete, given his debilitating back problem for which he needed spinal fusion surgery.
"To be just able to compete again, if you had said that last year at this particular time I would have said you're crazy, " he said.
Phil Mickelson also showed the form that he made him among the pre-tournament favourites as he carded a five-under 67, but having started the day at seven-over, it was mere consolation.
Shot of the week was reserved for the last day as American Charlie Hoffman as he aced the par-3 16th for the first hole-in-one of the tournament.
It was the 20th ace in Masters history at the 170-yard hole as his six-iron narrowly missed a bunker before rolling into the hole.
Another notable final round performance came from Tony Finau, who finished with a six-under 66 for seven-under 281, the same mark as world number one Dustin Johnson.
Finau dislocated his ankle in the Par-3 contest Wednesday while celebrating a hole-in-one and made a remarkable recovery to play in his first Masters, mounting a strong challenge on the first day despite his discomfort.
Reed's victory, succeeding Sergio Garcia, whose title defense was derailed by a disastrous quadruple bogey 13 on the first day, means that all four major golf titles are held by Americans of 27 years of age or under.
Brooks Koepka will defend his US Open crown in June, with Spieth holding the British Open title while world number two Justin Thomas is the USPGA champion.
Of a footnote, Reed did fail in his bid to become the first man in 82 Masters events to break 70 for all four days, but he will care little after forging his place in golfing history with such a resilient display, marked by superb putting under pressure.