Bradley, the nephew of LPGA Tour great Pat Bradley, sank a 2-foot par putt at the 419-yard 18th hole in the playoff. Palmer's approach went into the water and he made a bogey with a 13-foot putt.
About an hour earlier, Bradley finished his closing round of 2-under 68 with a par at No. 18, dropping into a squat and hopping a few times in frustration when his 10-foot birdie chance slid by the hole.
Palmer (72) and Bradley finished at 3-under 277, the highest winning score on the PGA Tour this year. It was the fifth playoff in six weeks.
"I don't know what to say. I'm so happy, I can't believe this just happened," Bradley said. "This is a dream come true. I've waited for this my whole life."
The PGA Tour rookie's aunt won 31 tournaments, including six majors. The first victory for the Vermont native who played at St. John's came nine days before his 25th birthday.
On the 172-yard 17th hole, Bradley sank a 12-foot par-saving putt and responded with an emphatic fist pump. After Bradley tapped in his par putt at No. 18, third-round leader Palmer was in one of the five groups still playing.
Bradley then sat for a few minutes before going to the practice range to prepare for a playoff that almost wasn't necessary for him to become the PGA Tour's sixth first-time winning this season.
Ryuji Imada (71) and Joe Ogilvie (70) finished a stroke back at 2 under. Imada bogeyed three of his last four holes after getting to 5 under.
Defending Nelson champion Jason Day (67) was fifth at 1 under. Only five players finished under par this week at TPC Four Seasons. There were brutal scoring conditions the last three days, particularly for both weekend rounds when the wind was sustained at 25 mph with gusts howling near 40.
It was the highest finishing score for the Nelson since the tournament moved to its current location in 1983.
After Palmer blasted from a greenside bunker to 3 feet for a birdie at the 523-yard 16th hole to get back to 3 under, he gave that stroke right back when he missed a 7-foot par putt at No. 17.
Palmer forced the playoff with a 6-foot birdie putt at No. 18, where there was only one other birdie all day. When the putt dropped, Palmer punched his right fist in the air and then raised both arms over his head.
Bradley and Palmer then played No. 18 again, both going way right with their tee shots to start the playoff.
Tournament volunteers quickly dismantled and moved a temporary lemonade stand to give Bradley a line of sight to the green and avoid the necessity for a drop.
Bradley's approach was dangerously close to sliding off the side of the green into the water, but stayed up. Palmer went in the same direction but his ball didn't stay dry.