Exclusive diffractive meson production represents a golden channel for investigating gluonic saturation inside nucleons and nuclei. In this Letter, we settle a systematic framework to deal with beyond leading power corrections at small x, including the saturation regime, and obtain the γ^{*}→M(ρ,ϕ,ω) impact factor with both incoming photon and outgoing meson carrying arbitrary polarizations. This is of particular interest since the saturation scale at modern colliders, although entering a perturbative regime, is not large enough to prevent higher-twist effects to be sizable.