The South Korean-born New Zealander earned $1.2 million and became the ninth first-time winner on the tour this season.
Lee, Canadian David Hearn, Kevin Kisner and Robert Streb were tied at 13 under after four rounds on the Old White TPC course in West Virginia.
Streb shot a 5-under 65 in the final round by making five birdie putts on the back nine with a 56-degree wedge after breaking his putter on the ninth hole. The putter broke when Streb tossed it at his bag next to the green.
He was allowed a replacement putter for the playoff, but he never got to use it after missing the green at the par-3 No. 18. Lee and Hearn made birdie putts on the first playoff hole, also eliminating Kisner.
Hearn, from Brantford, Ont., then drove behind a tree on the par-5 17th and made bogey. Lee reached the green in three shots and two-putted from across the green for par.
It marked the third playoff in the tournament's six-year history. A week ago, Bubba Watson won the Travelers Championship in a playoff.
Tiger Woods started the day seven shots behind the leaders, shot a bogey-free 67 and said he made some "nice strides" heading into next week's British Open.
Woods broke a streak of 55 consecutive rounds with at least one bogey. It was his first time under par in a final round since the 2013 Tour Championship.
Lee, Hearn, James Hahn and Greg Owen earned spots at St. Andrews. There also were four spots handed out a week ago at the Travelers and one more is available this week at the John Deere Classic.
Kisner shot a 6-under 64 to get to the clubhouse at 13 under. But he had to sweat it out with eight golfers within two shots of the lead still on the course.
Kisner lost in a playoff for the third time this season. The others were at the RBC Heritage and the Players Championship.
Kisner birdied the 18th three times in the first four rounds, but he went over the green in the playoff and left his approach shot in the rough.
Hearn (67) and Lee (67) birdied No. 17 to join the playoff. Both had chances to take the lead on the final hole in regulation, but Lee missed an 18-footer for birdie and Hearn left a 12-footer just short of the cup.
Russell Henley shot 63 and finished fifth at 12 under. Seven others were at 11 under, including Chad Collins (69), who was tied for the lead before bogeying the final two holes.
Jason Bohn, Sean O'Hair, Sun Joon Park and Bryce Molder shared the third-round lead at 11 under, but couldn't keep the momentum going Sunday. Molder shot 70 and the others had over-par rounds.