Neutrino-nucleus scattering cross sections are critical theoretical inputs for long-baseline neutrino oscillation experiments. However, robust modeling of these cross sections remains challenging. For a simple but physically motivated toy model of the DUNE experiment, we demonstrate that an accurate neural-network model of the cross section -- leveraging only Standard-Model symmetries -- can be learned from near-detector data. We perform a neutrino oscillation analysis with simulated far-detector events, finding that oscillation analysis results enabled by our data-driven cross-section model approach the theoretical limit achievable with perfect prior knowledge of the cross section. We further quantify the effects of flux shape and detector resolution uncertainties as well as systematics from cross-section mismodeling. This proof-of-principle study highlights the potential of future neutrino near-detector datasets and data-driven cross-section models.