The Frenchman recorded a double-bogey six on the second hole, but recovered to notch birdies on three of the last four holes en route to a round of 5-under 67.
Quesne's only other victory came at the 2012 Open de Andalucia Costa del Sol.
Sharing fourth place at 10-under were 2010 winner Fredrik Andersson Hed (67), Emiliano Grillo (68), Felipe Aguilar (71) and Nicolas Colsaerts (72).
Overnight leader Marcus Fraser was contending for the title for much of the final round before stumbling down the stretch to finish with a share of eighth place at 9-under 279.
Swim Week Miami Beach 2014 [PHOTOS]