The 28-year-old left-hander had missed five straight cuts entering the Greenbrier Classic and his previous best finish was a tie for 13th.
"When you're missing cuts every week, you get down on yourself," Potter said. "I mean, it's hard to pick yourself back up. But the plus side for me is I was still young. I just knew I had plenty of time and just be patient and it will come back around again."
During the fourth round, Potter made long putts for a birdie at No. 15 and an eagle at No. 17, and his 5-footer for birdie at No. 18 tied Kelly, who could have avoided the playoff but missed birdie putts on the final two holes.
Webb Simpson lost a one-stroke lead on the back nine. The U.S. Open champion made three straight bogeys, shot 73 and tied for seventh at 11 under.
Farmington's Daniel Summerhays, a former BYU golfer, finished fifth at 13 under after a 64.
Na Yeon Choi survived a triple bogey and a few more shaky moments on the back nine to win the U.S. Women's Open at Blackwolf Run in Kohler, Wis. It's the first major and sixth career LPGA Tour victory for the 24-year-old South Korean star, who came into the tournament ranked fifth in the world.
Choi shot a 1-over 73 and finished at 7 under for a four-stroke victory. Fellow South Korean player Amy Yang had a 71 to finish second. They were the only players to finish the tournament under par.
Choi came into Sunday with a six-stroke lead after shooting a 7-under 65 on Saturday. She got into trouble when she triple-bogeyed No. 10, but recovered to win at the same course where Se Ri Pak won South Korea's first major title in 1998. Pak was among a group of friends who met Choi after she putted out on the 18th green, showering her with hugs and victory champagne.
Kirk Triplett won the First Tee Open at Pebble Beach for his first Champions Tour title, making an early eagle and closing with a 6-under 66 for a two-stroke victory over Mark McNulty.