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Full manipulation of a quantum system requires controlled evolution generated by nonlinear interactions, which is coherent when the rate of nonlinearity is large compared with the rate of decoherence. As a result, engineered quantum systems typically rely on a bare nonlinearity much stronger than decoherence rates, and this hierarchy is usually assumed to be necessary. Here we challenge this assumption by demonstrating the universal control of a quantum system where the rate of bare nonlinear interaction is comparable to the fastest rate of decoherence. We introduce a noise-resilient protocol for the universal quantum control of a nearly harmonic oscillator that takes advantage of an in situ enhanced nonlinearity instead of harnessing a bare nonlinearity. Our experiment consists of a high-quality-factor microwave cavity with weak dispersive coupling to a superconducting qubit with much lower quality. By using strong drives to temporarily excite the oscillator, we realize an amplified three-wave-mixing interaction, achieving typical operation speeds over an order of magnitude faster than expected from the bare dispersive coupling. Our demonstrations include the preparation of a single-photon state with high fidelity, the generation of squeezed vacuum with large intracavity squeezing and measurement-free preparation of logical states for the binomial and Gottesman–Kitaev–Preskill quantum error-correcting codes.