Huh fired an eight-under 63 in the final round to get into the clubhouse at 13-under-par 271 at El Camaleon Golf Course.
Allenby, who hasn't won on the PGA Tour since the 2001 Pennsylvania Classic, was five groups behind Huh and in the lead at 15-under. However, he faltered to a double-bogey on the par-four 18th to close out a six-under 65.
The closing double-bogey left Allenby tied for the lead with Huh.
Chris Stroud, who was in the final group, was at 12-under with two holes to play and could have tied or passed the duo, but failed to do so.
In the playoff, the Huh and Allenby matched pars on the first seven holes. On the eighth, Allenby blinked. He tripped to a bogey, while Huh made his eighth par to secure his first PGA Tour title.
Matt Every, who shared sixth place at the Sony Open in Hawaii, and Colt Knost both closed with five-under 66s to share third place.
Third-round leader Daniel Summerhays stumbled to five bogeys on the front nine, but fought back to shoot two-over 73 that left him tied for fifth with Dicky Pride (66), Alejandro Canizares (66) and Stroud (71).
Stroud bogeyed the 16th, but still had a chance to get into the playoff with a birdie on one of the final two holes. He parred 17 but, like Allenby, faltered to a double-bogey at the last.
MORE TO FOLLOW.