Goydos closed with a 3-under 67 and made birdie on the last hole when his 25-foot chip banged into the pin and settled within tap-in range. Charles Howell III and Luke Donald tied for second, a stroke back.
"I never felt like I was going to win," said Goydos, who earned $936,000, more than he made all last year.
The tournament belonged to Howell for most of a sunny afternoon at Waialae until a sudden shift on the back nine, when Howell made back-to-back bogeys and Goydos made consecutive birdies.
Howell had a chance to force a playoff when his 5-iron from the rough came up 50 feet short of the green. His chip ran 15 feet past the pin, and the birdie putt never had a chance. He shot 70 for his seventh runner-up since his only victory in 2002.
Donald had an eagle chip on the par-5 18th that would have forced a playoff, but it hit the pin and spun away, leaving him with a 69.
Tadd Fujikawa, the 16-year-old who became the youngest player in 50 years to make a cut on the PGA Tour, finished his dream week with a birdie on the final hole for a 72, putting him in a tie for 20th.
"I never imagined myself doing this, especially at this age," Fujikawa said, who returns to the 10th grade on Tuesday.