We present NuSTAR high-energy X-ray observations of the pulsar wind nebula (PWN)/supernova remnant G21.5-0.9. We detect integrated emission from the nebula up to similar to 40 keV, and resolve individual spatial features over a broad X-ray band for the first time. The morphology seen by NuSTAR agrees well with that seen by XMM-Newton and Chandra below 10 keV. At high energies, NuSTAR clearly detects non-thermal emission up to similar to 20 keV that extends along the eastern and northern rim of the supernova shell. The broadband images clearly demonstrate that X-ray emission from the North Spur and Eastern Limb results predominantly from non-thermal processes. We detect a break in the spatially integrated X-ray spectrum at similar to 9 keV that cannot be reproduced by current spectral energy distribution models, implying either a more complex electron injection spectrum or an additional process such as diffusion compared to what has been considered in previous work. We use spatially resolved maps to derive an energy-dependent cooling length scale, L(E) proportional to E-m with m = -0.21 +/- 0.01. We find this to be inconsistent with the model for the morphological evolution with energy described by Kennel & Coroniti. This value, along with the observed steepening in power-law index between radio and X-ray, can be quantitatively explained as an energy-loss spectral break in the simple scaling model of Reynolds, assuming particle advection dominates over diffusion. This interpretation requires a substantial departure from spherical magnetohydrodynamic, magnetic-flux-conserving outflow, most plausibly in the form of turbulent magnetic-field amplification.