We present our discovery of a narrow-line Baldwin effect, an anticorrelation between the equivalent width (EW) of a line and the flux of the associated continuum, in 5-20 μm mid-infrared (mid-IR) lines from a sample of 68 active galactic nuclei (AGNs), located at z< 0.5, observed with the Infrared Spectrograph on the Spitzer Space Telescope. Our analysis reveals a clear anticorrelation between the EW of the [S IV] 10.51 μm, [Ne II] 12.81 μm, and [Ne III] 15.56 μm lines and their mid-IR continuum luminosities, while the Baldwin effect for [Ne V] 14.32 μm is not as obvious. We suggest that this anticorrelation is driven by the central AGN, and not circumnuclear star formation in the host galaxy, and present a new method of analyzing this effect in mid-IR lines. We also find that the slope of the narrow-line Baldwin effect in the mid-IR does not appear to steepen with increasing ionization potential. Examining the dependence of the EW on the Eddington ratio (L/L _Edd), we find no strong relationship for mid-IR lines. Our study indicates that the narrow-line mid-IR Baldwin effect is quite different from the broad-line optical/UV Baldwin effect, and it is possible that the two effects are unrelated. The anticorrelations discovered open new possibilities in understanding the physics of the ionizing region and the continuum reprocessing by dust.