Observations of some quiescent black hole X-ray binaries have revealed an excess of mid-infrared (MIR) emission above that expected from their donor stars. In one system, V404 Cygni, this excess has been variously suggested to arise from the accretion disk, circumbinary material, or a compact relativistic jet. Here we present simultaneous James Webb Space Telescope (JWST), Atacama Large Millimeter/submillimeter Array, and complementary multiwavelength observations undertaken to resolve this uncertainty. We observed large-amplitude 21 μm variability on short timescales with JWST, particularly a dramatic flare, which swiftly rose to ≈2.4 mJy, over 10 times the lowest observed MIR flux density. Similar variability was simultaneously observed from radio to X-ray wavelengths with other facilities throughout the campaign. This variability and the flat radio/millimeter/MIR spectral index (α = 0.04 ± 0.01) suggest that the MIR excess at and longward of 21 μm in V404 Cyg does not arise from the accretion disk or circumbinary material. Instead, the emission at 21 μm is dominated by synchrotron radiation from a jet, which persists into quiescence. This result reinforces the ubiquity of the disk–jet connection in accreting black holes across a range of masses and accretion rates.