The 29-year-old American, who won the Masters in 2018, began the day one shot ahead of Abraham Ancer and that is the way it finished after the Mexican also carded a two-under-par 69.
Reed made three birdies and three bogeys on his outward nine holes, but did not drop another shot and picked up two more birdies at the 14th and 16th.
Ancer bogeyed the fourth and sixth but shot four birdies in total, at the third, eighth, 16th and 17th, to finish one stroke ahead of American Harold Varner III and Spain’s Jon Rahm, who shot 68 and 69 respectively.
Australian Adam Scott posted the final round’s joint lowest total of 65, with his seven birdies and a bogey lifting him up 12 places to fifth.
Rory McIlroy, Justin Rose and Danny Willett all started the day amongst the chasing pack but were unable to make any impression on the top two.
McIlroy, four shots behind overnight, birdied seven holes but dropped shots at five others to finish with a 69, still four shots adrift and tied for sixth place alongside Jordan Spieth, Louis Oosthuizen and Brandt Snedeker.
Rose shot a level-par 71 to finish joint 10th with fellow countryman Ian Poulter, who closed with a 68, while Willett fell away to finish tied in 24th place after a four-over-par 75.