Crane did not make a single birdie in the final round, but a three-over-par 73 was enough to close the deal at TPC Southwind.
He had the luxury of bogeying the final hole to finish at 10-under 270, while compatriot Troy Merritt (71) claimed second place on nine-under.
Crane, 38, collected $1.044 million (£595,000) for his fifth PGA Tour victory.
He will have a week off to celebrate his triumph, because he is not in the field for the US Open which starts on Thursday.
Crane held a three-shot lead over Merritt after 54 holes.
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