MIAMI — Andrew Gillum’s campaign bus was running late, tied up in rain and traffic on Interstate 95, so his relatives waiting to catch a glimpse of the final days of his surging candidacy had a few minutes to reminisce about the politician as a young boy.

“He’s always been a kid that was different from all the rest,” George Jackson, his 71-year-old uncle, said of Mr. Gillum, the son of a bus driver and a construction laborer who grew up in the southern end of Miami-Dade County. “Always had a book up to his head. His grandmother always told him that he was special.”

Grandma, it seems, was right.

On Tuesday, Mr. Gillum, 39, became the first black nominee for Florida governor, achieving a stunning and improbable come-from-behind win over four wealthy Democratic challengers whose personal fortunes proved no match for Mr. Gillum’s compelling life story and progressive message.

“We’re going to unite this state,” Mr. Gillum, the mayor of Tallahassee, told a crowd of supporters on Tuesday night. “What’s going to bring us together is our common and shared belief that regardless of where you come from, regardless of what your mother or your father did for their profession, regardless of what side of the tracks you live on that, that every single Floridian ought to have their equal and fair shot at the American dream.”