The Duffing oscillator is a nonlinear extension of the ubiquitous harmonic oscillator and as such plays an outstanding role in science and technology. Experimentally, the system parameters are determined by a measurement of its response to an external excitation. When changing the amplitude or frequency of the external excitation, a sudden jump in the response function reveals the nonlinear dynamics prominently. However, this bistability leaves part of the full response function unobserved, which limits the precise measurement of the system parameters. Here, we exploit the often unknown fact that the response of a Duffing oscillator with nonlinear damping is a unique function of its phase. By actively stabilizing the oscillator's phase we map out the full response function. This phase control allows us to precisely determine the system parameters. Our results are particularly important for characterizing nanoscale resonators, where nonlinear effects are observed readily and which hold great promise for next generation of ultrasensitive force and mass measurements. We demonstrate our approach experimentally with an optically levitated particle in high vacuum.