Blixt came from four strokes down at the start of the final round to win the rain-delayed tournament by two strokes Sunday.
The Swede shot a 3-under 67 to finish at 13-under 267. Playing in the next-to-last group, he was overcome with emotion when the final pairing of third-round leader Johnson Wagner and Jimmy Walker came up short of the holes-in-one they needed at No. 18 to force a playoff.
Part of that reaction was due to knowing he'll be heading to Sweden to see his family for the first time in over a month.
"This kind of means that I can relax a little bit," Blixt said. "It's a big hunt during the year trying to get into the FedEx Cup playoffs.
"Not that the hunt is over, but it means that maybe I can take a week off and go back and see them and make a little bit more time for them instead of looking forward to seeing them at Christmas."
The win boosted Blixt from 139th to 39th in the FedEx Cup points standings. The top 125 players will make the playoffs starting in August.
Of course, there are other perks, including a spot in next month's PGA Championship and 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.
Blixt emerged from a five-player chase over the final five holes to pick up the $1.1 million winner's check.
Wagner (73), Walker (71) and Australians Steven Bowditch (68) and Matt Jones (68) tied for second at 11 under.
Blixt adds to his win at the Frys.com Open as a rookie last year.
"My coach, when I was a kid, used to say, 'Once is luck, twice is skill,'" he said.
Entering the Greenbrier Classic, Blixt didn't have a top-10 finish this season, missing as many cuts as he made.
"It's just been a hard year," he said. "My game has not been on."
It was Sunday.
Blixt made five birdies, including a 9-footer at No. 16 that moved him into the lead for good at 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.
Play on the Old White TPC course was halted for three hours due to thunderstorms. The last group teed off at 5:10 p.m. EDT and finished just after sunset. The tour narrowly avoided going past a Sunday finish for the fourth time this year.
"It was really dark," Wagner said.
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."
Defending champion Ted Potter Jr. (67), Pat Perez (69) and Brian Stuard (67) tied for sixth at 9 under. Among those in an eight-way tie for ninth place was Davis Love III, who earned his first top-10 finish of the year.
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 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 71 Sunday to finish in a tie for 62nd at even par.
"It was a blast," Palmer said. "I was more relaxed. A vacation golf tournament. I wish I could have played better."