A novel theoretical non-Hermitian formalism for analyzing nonlinear nano-optics is proposed. Its main strength lies in the unique way it analytically incorporates the quasinormal modes of the nanostructures, which have to be resonantly excited to achieve significant nonlinear efficiency. Owing to the analyticity, the formalism is computationally more effective than the usual multipolar Mie-scattering expansions for nonlinear nano-optics. It is also more general and applies to nanostructures laying on substrates or embedded in thin films. It additionally provides guidelines for the multiparameter design and optimization of nonlinear photonic nanostructures. In particular, it reveals an important phase-matching condition at the subwavelength scale between the linear and nonlinear harmonics, which was not clarified in earlier theoretical works based on Hermitian theory for waveguides and high-Q cavities. Its closed form is a major asset that enables us to propose a systematic approach to design phased-matched nanostructures offering drastic second harmonic generation enhancements with engineered χ(2) and pump beams.