The Swede shot a 3-under 67 Sunday to win the rain-delayed tournament by two strokes.
Blixt emerged from a five-player chase over the final five holes to pick up the $1.1 million winner's check. He also shot from 139th to 39th in the FedEx Cup points standings.
''This is what I play for,'' Blixt said. ''I play to win. It just confirms that if you do the right things, that you work hard, dreams can come true.''
Among the perks for his victory is a spot in next year's Masters.
Blixt also will move to around No. 50 in the next world ranking, which is used as the alternate list to fill the field for this month's British Open. That would make him the top alternate in a year that about eight players from the list will get into the Open at Muirfield.
He overcame a four-shot deficit at the start of the final round and finished at 13-under 267. Third-round leader Johnson Wagner (73), Australians Steven Bowditch (68) and Matt Jones (68), and Jimmy Walker (71) tied for second at 11 under.
Blixt went from a tie to a two-shot lead when he made a 9-foot birdie putt on No. 16 to move to 13 under. No other player made a birdie after that. Wagner bogeyed the par-3 15th moments later to fall to 11 under alongside Bowditch and Walker.
Blixt also won the Frys.com Open last year as a tour rookie. But entering the Greenbrier Classic, he hadn't had a top-10 finish this season, missing as many cuts as he made.
Blixt was overcome with emotion after watching Wagner and Walker, needing holes-in-one at No. 18, reach the green but land well away from the hole.
''It's just been a hard year,'' Blixt said. ''My game has not been on.''
Wagner, who had missed out on weekend play in his last seven tournaments, couldn't match the seven birdies he had in the third round on his way to a 64. He bogeyed three holes in a five-hole stretch on the back nine and never recovered.
The 54-hole leader has yet to win the Greenbrier Classic, now in its fourth year.
''The swing just left,'' Wagner said. ''I'm furious. But given where I was a couple of weeks ago, I'll take a lot of positives when I get over this disappointment right now.''
While Bowditch couldn't make up a five-shot deficit, he earned his first top 10 since Pebble Beach in 2011.
Play on the Old White TPC course was halted for three hours because of thunderstorms. The last group teed off at 5:10 p.m. ET and finished just after sunset. The Tour narrowly avoided going past a Sunday finish for the fourth time this year.
Spieth ran off nine straight pars to start Sunday's round, then had two bogeys in a four-hole stretch. He shot 73 and finished tied for 23rd at 6 under. The 19-year-old Texan is still searching for an elusive win that would give him his PGA Tour membership and make him eligible for the FedEx Cup playoffs. He is assured of a Tour card when the new season starts in October.
Walker joined four others after the Greenbrier Classic to earn spots in the British Open as the leading five players not already exempt from the top 20 in the FedEx Cup standings. He moved up to 17th. The others are Billy Horschel, Boo Weekley, Russell Henley and Harris English.
The Greenbrier Classic has had close finishes in every year of its four-year existence. The tournament went to playoffs the past two years and Stuart Appleby shot 59 to win by a stroke in 2010.
The tournament's end means New Orleans Saints coach Sean Payton can go back to his regular job after carrying the bag of friend Ryan Palmer.
''It's a good grind, good exercise,'' Payton said. ''I thought Ryan was really patient.''
Palmer shot a 70 Sunday to finish in a tie for 59th at 1 under.
''It was a blast,'' Palmer said. ''I was more relaxed. A vacation golf tournament. I wish I could have played better.''