Feedback from central supermassive black holes is often invoked to explain the low star formation rates (SFRs) in the massive galaxies at the centers of galaxy clusters. However, the detailed physics of the coupling of the injected feedback energy with the intracluster medium (ICM) is still unclear. Using high-resolution magnetohydrodynamic cosmological simulations of galaxy cluster formation, we investigate the role of anisotropic thermal conduction in shaping the thermodynamic structure of clusters, and in particular, in modifying the impact of black hole feedback. Stratified anisotropically conducting plasmas are formally always unstable, and thus more prone to mixing, an expectation borne out by our results. The increased mixing efficiently isotropizes the injected feedback energy, which in turn significantly improves the coupling between the feedback energy and the ICM. This facilitates an earlier disruption of the cool-core, reduces the SFR by more than an order of magnitude, and results in earlier quenching despite an overall lower amount of feedback energy injected into the cluster core. With conduction, the metallicity gradients and dispersions are lowered, aligning them better with observational constraints. These results highlight the important role of thermal conduction in establishing and maintaining the quiescence of massive galaxies.