Huh, who had never competed on the U.S. circuit until this year, sealed victory at the eighth extra hole to deny the 40-year-old Allenby a first PGA Tour win since the 2001 Pennsylvania Classic.
The two players had finished the regulation 72 holes level at 13-under 271, Huh closing with a flawless eight-under 63 and Allenby double-bogeying the last for a 65 on the El Camaleon course.
"It's amazing, I can't believe it happened right now," Huh, 21, told reporters. "It was my dream to play on the PGA Tour and now, finally winning a tournament. It's awesome.
"Playing an eight-hole playoff is long, and I was definitely nervous out there, trying to calm myself down. It was hard," added Huh, who came from seven shots off the lead going into the final round.
Sunday's playoff matched the second longest on the PGA Tour. The 1949 Motor City Open in Michigan ended deadlocked after 11 playoff holes when Cary Middlecoff and Lloyd Mangrum opted to split the purse by mutual agreement.
"I had this tournament in the bag, a two-shot lead with one hole to play and just played it like a rookie, pretty much," he said. "I had played so awesome all day.
"I putted great, I hit the ball great, and I did everything that I needed to do coming down the stretch and then to give it away at 18 is very, very disappointing."
Allenby, whose haul of eight birdies in the final round included five in a row from the par-four second, applauded the winning effort by Huh, who describes himself as a Korean-American.
"John's a great player," he said. "I played with him yesterday, and he's got a great future. He's a young guy, and all the best to him. It's great to see him win."
Americans Matt Every and Colt Knost signed off with matching 66s to share third place at 11 under.
(Writing by Mark Lamport-Stokes in Marana, Arizona; Editing by Nick Mulvenney)