A Legendre--Fenchel transform framework is developed for pressure-driven capillary flow of monotone generalized Newtonian fluids. A monotone nonlinear flow curve defines a convex dissipation potential, and its Legendre--Fenchel transform gives a complementary potential in the stress variable. Since the stress distribution is known explicitly in fully developed capillary flow, the velocity profile can be obtained from this complementary potential without requiring the inverse constitutive relation to be written explicitly. This reformulates nonlinear capillary-flow modeling from a model-by-model constitutive inversion problem into a convex-duality construction. The method recovers the shear-rate-parametrized Carreau--Yasuda solution, gives compact average-velocity relations for circular and planar geometries, and extends to several generalized Newtonian models, including Papanastasiou-regularized yield-stress fluids. In the ideal yield-stress limit, plug formation follows naturally from the subdifferential structure of the dissipation potential and the corresponding flat interval of the complementary potential. The framework provides a unified mathematical structure for analytical and semi-analytical pressure-drop--flow-rate relations in nonlinear capillary flow.