Lee started the day at Memorial Park leading by four shots and still led by three on 21 under heading into the final three holes, but saw his advantage shrink after finding the water on the par-five 16th and carding a bogey.
Meanwhile, Gary Woodland had surged into contention with an eagle and two birdies in his final four holes, matching the course record with a 62 to set the clubhouse lead at 19 under.
World number one Scottie Scheffler, playing in the penultimate group, had a chance to join the lead on the 17th but missed a putt for a fifth consecutive birdie. He closed with a 63, also finishing at 19 under.
Needing a par on the 18th to seal the win, Lee held his nerve, signing off with a 67 to finish one shot ahead at 20 under.
“It was hard, really hard,” said Lee. “Scottie is a wonderful golfer and he keeps you on your toes.
“This is my first time being in front and trying to hold a lead. I’m glad I got it done, but man, I’m just very exhausted.
“It was a lot of mental grind. I’m so proud of the way I handled myself.”
Northern Ireland’s Rory McIlroy closed with a superb six-under-par 64, featuring an eagle, six birdies, and a bogey, to reach 15 under and tie for fifth place.
“It was a good week,” said McIlroy, who will be aiming to finally complete a career Grand Slam by winning the Masters at Augusta next month.
“I still feel like I’ve got some stuff to work on. Overall, a solid week and nice to have another week to get ready and fine tune my game for Augusta.
“My right elbow has been bothering me a little bit so [I’ll] maybe just get some treatment on that to make sure it’s OK going into Augusta.”
