We present JWST/MIRI mid-infrared integral field spectroscopy combined with ALMA CO(2--1) observations of the post-starburst galaxy ESO 420-G13, hosting a low-luminosity active galactic nucleus (AGN). The unprecedented spatial and spectral resolution of MIRI enables a detailed study of the molecular and ionised gas kinematics, excitation, and energetics in the nuclear kiloparsec, revealing the impact of AGN feedback in a system with modest radiative output. Despite its faint radio and X-ray emission (L2--10 keV ~ 1040 erg s-1), ESO 420-G13 exhibits powerful kinetic feedback in the form of massive molecular and ionised gas outflows, with a total kinetic power of ~1.4 × 1041 erg s-1. This corresponds to a jet--interstellar medium coupling efficiency of ~3.8%, within the range observed in more powerful AGNs. The feedback is driven by a previously undetected compact jet, traced by collimated coronal-line and extended X-ray emission to ≳870 pc from the nucleus. The interaction is strongest ~370 pc north of the nucleus, where a fast ionised gas stream emerges perpendicular to the jet axis, coinciding with a bend in the jet direction. Enhanced velocity dispersion in warm H2 surrounds this gas stream, consistent with an expanding molecular bubble. Massive molecular outflows are detected at its edges; the blueshifted outflow is devoid of CO emission, likely due to CO destruction in shocks or by cosmic rays from the jet--ISM interaction. About 5% of the central molecular reservoir has already been expelled, and the remaining gas is turbulent and warm, suggesting an ongoing phase of AGN-driven feedback in this post-starburst galaxy. Our results highlight the enormous potential of mid-infrared imaging spectroscopy to uncover jet-driven feedback in low-luminosity AGNs. Without the spatially resolved MIRI diagnostics, the kinetic power of the AGN in ESO 420-G13 and its role in shaping the host galaxy ISM would have remained hidden.