To determine the nature of dark matter (DM) using astrophysical probes, the next milestone is to constrain the properties of structure on subgalactic scales. Gravitational lensing offers a remarkably powerful probe of this regime: in particular, statistics of flux-ratio anomalies (discrepancies between mass model predictions and observed flux ratios) in quadruply imaged quasars are sensitive to perturbations by low-mass DM halos down to $\sim {10}^6{M}_{\odot }$, independent of their baryonic content. Studies leveraging these anomalies require high-quality flux-ratio measurements from an emission region insensitive to stellar microlensing. In this paper, we present the first measurement of narrow-line flux ratios for a gravitationally lensed quasar using JWST/NIRSpec with Integral Field Spectroscopy (IFS), targeting the well-studied system RXJ1131-1231. Flux ratios are extracted from the [S III] 9071/9533 Å narrow-line doublet — the first use of this doublet for substructure studies — by performing a full lens model reconstruction to isolate the unresolved nuclear emission from extended narrow-line emission. The resulting spectra are jointly modeled using , a new, publicly available software package introduced in this work for the simultaneous spectral fitting of multiple lensed quasar images. We achieve ~5% uncertainties on the flux ratios, comparable to the precision of JWST/MIRI warm dust measurements, and detect a clear anomaly in the cusp images relative to a standard smooth lens model. Our results are in good agreement with previous narrow-line measurements and broadly consistent with JWST/MIRI warm dust flux ratios, with marginal (~2-3 $\sigma$) deviations. We illustrate how such systematic shifts between differently sized emission regions may be enhanced by small spatial offsets of the order of tens of parsecs in the presence of a low-mass dark matter halo. Our method is generalizable to other systems with existing or future IFS observations, and the combination of narrow-line and warm dust flux ratios offers a new avenue for improving DM substructure constraints with flux-ratio anomaly statistics.