The inverse-Compton effect (IC) is a widely recognized cooling mechanism for both relativistic and thermal electrons in various astrophysical environments, including the intergalactic medium and X-ray emitting plasmas. Its effect on thermal electrons is however frequently overlooked in theoretical and numerical models of colliding-wind binaries (CWB). In this article, we provide a comprehensive investigation of the impact of IC cooling in CWBs, presenting general results for when the photon fields of the stars dominate the cooling of the thermal plasma and when shocks at the stagnation point are expected to be radiative. Our analysis shows that IC cooling is the primary cooling process for the shocked-wind layer over a significant portion of the relevant parameter space, particularly in eccentric systems with large wind-momentum ratios, e.g., those containing a Wolf-Rayet and O-type star. Using the binary system WR 140 as a case study, we demonstrate that IC cooling leads to a strongly radiative shocked wind near periastron, which may otherwise remain adiabatic if only collisional cooling was considered. Our results are further supported by 2D and 3D simulations, which illustrate the impact of including or neglecting IC cooling. Specifically, 3D magnetohydrodynamic simulations of WR 140 show a significant decrease in hard-X-ray emission around periastron, in agreement with observations but in contrast to equivalent simulations that omit IC cooling. A novel method is proposed for constraining mass-loss rates of both stars in eccentric binaries where the wind-collision zone switches from adiabatic to radiative approaching periastron. This study highlights the importance of including the IC process for thermal electrons in theoretical and numerical models of colliding-wind binaries.