Goydos' 28-foot putt from off the green on the 16th highlighted a third-round 63, tying Hunter Mahan for the best round of the day.
Perry, who led after each of the first two rounds, had a seemingly comfortable three-stroke lead after birdies on 10, 11 and 12. But he dropped a stroke on the 14th and Goydos took advantage, making four straight birdies starting on the par-5 13th.
After a rain delay of just over an hour, both golfers parred in. Perry missed a 5-foot birdie try on 18 that would have given him a share of the lead minutes after Goydos misread the break on a 10-footer.