The ability to predict and control the formation of bimetallic heterogeneous nanocrystals is desirable for many applications in plasmonics and catalysis. Here, we report the synthesis and characterization of stable, monodisperse, and solution-processed Cu-Ag bimetallic nanoparticles with specific but unusual elemental arrangements that are consistent with a recently developed thermodynamic model. Using air-free scanning transmission electron microscopy with energy-dispersive X-ray spectroscopy, the distribution of Cu and Ag positions was unambiguously identified within individual nanocrystals (NCs), leading to the discovery of a Cu-Ag nanocrescent shape. A simple yet versatile thermodynamic model was applied to illustrate how the interplay between surface and interface energies determines the particle morphology. It is found that there exists a range of surface-to-interface energy ratios under which crescent-shaped nanocrystals are the thermodynamically favored products, with the morphology tunable by adjusting the Ag content. We further show the conversion of Cu-Ag nanocrescents into Ag@Cu2O upon mild oxidation, whereas fully core-shell Cu@Ag NCs are robust against oxidation up to 100 degrees C. The plasmonic and interband absorptions of Cu-Ag NCs depend on the composition and the degree of Cu oxidation, which may find application in light-driven catalysis.