The periodicity inherent to any interferometric signal entails a fundamental trade-off between sensitivity and dynamic range of interferometry-based sensors. Here, we develop a methodology for substantially extending the dynamic range of such sensors without compromising their sensitivity, stability, and bandwidth. The scheme is based on simultaneous operation of two nearly identical interferometers, providing a moiré-like period much larger than 2π and benefiting from close-to-maximal sensitivity and from suppression of common-mode noise. The methodology is highly suited to atom interferometers, which offer record sensitivities in measuring gravito-inertial forces but suffer from limited dynamic range. We experimentally demonstrate an atom interferometer with a dynamic-range enhancement of more than an order of magnitude in a single shot and more than three orders of magnitude within a few shots for both static and dynamic signals. This approach can considerably improve the operation of interferometric sensors in challenging, uncertain, or rapidly varying conditions.