The X-ray binary system Cygnus X-3 (4U 2030+40, V1521 Cyg) is luminous but enigmatic owing to the high intervening absorption. High-resolution X-ray spectroscopy uniquely probes the dynamics of the photoionized gas in the system. In this Letter, we report on an observation of Cyg X-3 with the XRISM/Resolve spectrometer, which provides unprecedented spectral resolution and sensitivity in the 2–10 keV band. We detect multiple kinematic and ionization components in absorption and emission whose superposition leads to complex line profiles, including strong P Cygni profiles on resonance lines. The prominent Fe xxv Heα and Fe xxvi Lyα emission complexes are clearly resolved into their characteristic fine-structure transitions. Self-consistent photoionization modeling allows us to disentangle the absorption and emission components and measure the Doppler velocity of these components as a function of binary orbital phase. We find a significantly higher velocity amplitude for the emission lines than for the absorption lines. The absorption lines generally appear blueshifted by ∼−500–600 km s⁻¹. We show that the wind decomposes naturally into a relatively smooth and large-scale component, perhaps associated with the background wind itself, plus a turbulent, denser structure located close to the compact object in its orbit.