The James Webb Space Telescope (JWST) has opened up a new window to study highly reddened explosive transients. We present results from late-time JWST follow-up spectroscopic observations with NIRSpec and MIRI-LRS of the intermediate-luminosity red transient (ILRT) AT 2019abn. ILRTs represent a mysterious class of transients that exhibit peak luminosities between those of classical novae and supernovae and that are known to be highly dust obscured. Similar to the prototypical examples of this class of objects, NGC 300 2008-OT and SN 2008S, AT 2019abn has an extremely red and dusty progenitor detected only in pre-explosion Spitzer/IRAC imaging at 3.6 and 4.5 μm and not in deep optical or near-infrared Hubble Space Telescope images. We find that late-time observations of AT 2019abn from NEOWISE and JWST are consistent with the late-time evolution of SN 2008S. In part because they are so obscured by dust, it is unknown what produces an ILRT, with hypotheses including high-mass stellar merger events, nonterminal stellar outbursts, and terminal supernova explosions through electron capture in super-AGB (SAGB) stars. Our JWST observations show strong mid-IR class C polycyclic aromatic hydrocarbon features at 6.3 and 8.25 μm typical of carbon-rich post-AGB sources. These features suggest that the dust around AT 2019abn is composed of carbonaceous grains, which are not typically observed around red supergiants. However, depending on the strength and temperature of hot bottom burning, SAGB stars may be expected to exhibit a carbon-rich chemistry. Thus, our JWST observations are consistent with AT 2019abn having an SAGB progenitor and exploding as an electron-capture supernova.