Quantum theory promises sensors with dramatically improved precision relative to their classical counterparts, but noise quickly ruins these advantages. We demonstrate how a probe traversing noisy channels and a to-be-measured channel in a superposition of orders controlled by an auxiliary quantum system can withstand arbitrarily more noise than any quantum system with definite causal order, even if the probe is fully mixed and one of the noisy channels erases all information. After developing the theory of noise-resilient metrology with indefinite causal order, we discuss our experiments demonstrating the sensing of a unitary subject to depolarization noise using light's spatial and polarization degrees of freedom.