Humanity is close to characterizing the atmospheres of rocky exoplanets due to the advent of JWST. These astronomical observations motivate us to understand exoplanetary atmospheres to constrain habitability. We study the influence greenhouse gas supplement has on the atmosphere of TRAPPIST-1e, an Earth-like exoplanet, and Earth itself by analyzing ExoCAM and CMIP6 model simulations. We find an analogous relationship between CO2 supplement and amplified warming at non-irradiated regions (night side and polar)—such spatial heterogeneity results in significant global circulation changes. A dynamical systems framework provides additional insight into the vertical dynamics of the atmospheres. Indeed, we demonstrate that adding CO2 increases temporal stability near the surface and decreases stability at low pressures. Although Earth and TRAPPIST-1e take entirely different climate states, they share the relative response between climate dynamics and greenhouse gas supplements.