The 34-year-old Streelman shot a final round of four-under-par 67 to finish on 10-under, two strokes clear of his nearest rival at the tricky Copperhead layout at Innisbrook Resort.
"That was really cool," he said in a greenside interview with NBC television.
"It just shows that if you follow your dreams anything is possible."
Boo Weekley finished second after closing with a 63, the best single round of the tournament, to catapult himself up the leaderboard.
Weekley made eight birdies, including the 15th and 16th holes when he struck his approach shots to with two feet of the flagstick. He ate some pizza while he waited almost three hours for the later groups to finish.
"It was impressive. Even I'm still kind of shocked at how well I really hit it," Weekley said.
"It was one of the best ballstriking days I've had in a long time. I had three or four tap-ins today."
Cameron Tringale was third at minus seven after carding a 66 while Justin Leonard, the 1997 British Open champion, finished tied for fourth at minus six with last year's winner, Englishman Luke Donald, and Australian lefthander Greg Chalmers.
Streelman is anything but an overnight success.
He burnt out three cars driving himself around the United States as a struggling player trying to make the big time and worked as a caddie to pay his bills.
He joined the PGA Tour in 2008 and his previous best finish was third, which he had done three times.
Streelman started the final day in a three-way tie for the lead with Leonard and George Coetzee after a flawless round of 65 on Saturday.
He birdied two of his first three holes on Sunday and did not drop a shot all day, adding two birdies to secure his maiden Tour win and guarantee himself a start at Augusta.
"This is a dream come true. I think 153rd event on the Tour, and many, many events before that on the Hooters Tour and Gateway Tour and Dakotas Tour," he said at a news conference.
"To get this is the cumulation of a lot of hard work and a lot of time spent late into the evening and getting up early in the morning
"It's not going to change who I am. It's just something to put on a resume. It's really nothing more than that. But I'm very thankful and it means a lot to me and my team and people that I've surrounded myself with.
"Hopefully we can do it again. This is a lot of fun."
(Reporting by Julian Linden in New York; Editing by Gene Cherry)