Crane went three over in the final round, carding a 73, but still finished ahead of compatriot Troy Merritt on 10 under overall at TPC Southwind.
Merritt hit a 71 to end the tournament in sole possession of second spot on nine under overall.
Crane, who also had to complete the final 12 holes of his third round due to weather delays, clinched his first PGA Tour event since edging Webb Simpson in a play-off at the McGladrey Classic in 2011.
Simpson was among the contenders throughout the event as he climbed up the leaderboard courtesy of a round of 66 to finish in a tie for third at eight under.
The American was joined by counterpart Matt Every (70) and Swede Carl Pettersson (69).
A shot further back was Ian Poulter (64), who rounded out the top 10 alongside James Hahn (67), Andrew Svoboda (70), Brian Harman (72) and Billy Horschel (70).
Crane had to earn his fifth PGA Tour title on Sunday, after rain delayed the completion of the third round.
The 38-year-old was just six holes into the penultimate round on Saturday before play was suspended, but he returned on Sunday to retain his three-shot advantage.
Crane then quickly teed off at the first hole to open his final round, though it was far from ideal.
He made five successive pars before bogeys at the sixth and ninth holes threatened to derail his quest for the crown.
However, Crane steadied to sink eight consecutive par-putts on the back nine as he stayed clear of Merritt, who could not add to his two birdies on the front nine.
Meanwhile, American veteran Phil Mickelson (72) had four bogeys during his final round to finish 11th – four shots adrift of Crane.
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