He had the luxury of bogeying the final hole, tapping in from one foot to finish at 10-under 270, while fellow American Troy Merritt (71) claimed second place on nine-under.
Crane, 38, collected $1.044 million but the victory did not earn him a spot in the U.S. Open which starts on Thursday.
Still he was delighted to win again after changing his swing over the past year to help cope with a back problem.
"It's been a battle this year," he told CBS television. "I've changed a lot of stuff and I didn't know if I would get it back. This is a big surprise."
Crane, who led after every round, held a three-shot edge over Merritt after 54 holes and Merritt was the only player to exert any pressure in a final round devoid of drama.
Merritt, ranked 805th in the world, got within one stroke after nine holes, but did not pick up a birdie on the back nine.
Neither did Crane, who became the first player since Justin Leonard in 2005, also in Memphis, to win without a final round birdie.
Crane started Sunday perfectly with his first shot of the day, chipping in for birdie at the seventh hole in the weather-delayed third round.
That was his only birdie all day.
Phil Mickelson, meanwhile, tied for 11th, four strokes behind, matching his best result on the PGA Tour this year, though he tied for second on the European Tour in Abu Dhabi in January.
Mickelson needs to win the U.S. Open at Pinehurst, North Carolina, to complete the career Grand Slam of all four majors.
(Reporting by Andrew Both in Cary, North Carolina, editing by Gene Cherry)
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