The American led by two shots at Torrey Pines going down the 17th on the final day, but a bogey meant his lead was down to one with one hole to play.
However his playing partner, Australia's Michael Sim, could only par the last, meaning par on the 72nd hole was enough for Crane to win his third PGA Tour event with a final-round two-under par 70.
The 33-year-old began the day two shots adrift of overnight leader Ryuji Imada but a closing 75 put paid to the Japanese star's chances while Sim shared second place with Australian Marc Leishman, who shot 68 on Sunday and American Brandt Snedeker, who carded 69 for his final 18 holes.
Ernie Els finished a further shot back on -11, along with Michael Allen, Rickie Fowler and Alex Pugh.
World number two Phil Mickelson struggled in his home town on the final day, starting with three bogeys in a row to finish on eight-under after a one-over 73.
England's Justin Rose shot a two-under 70 to finish on -6, while Scotsman Martin Laird ended with a 68 to finish five-under and Luke Donald'3 73 saw him finish on one-under par.