We develop a simple yet comprehensive method to distinguish the underlying drivers of galaxy quenching, using the clustering and galaxy-galaxy lensing of red and blue galaxies in SDSS. Building on the iHOD framework developed by \citet{zm15}, we consider two quenching scenarios: 1) a “halo” quenching model in which halo mass is the sole driver for turning off star formation in both centrals and satellites; and 2) a “hybrid” quenching model in which the quenched fraction of galaxies depends on their stellar mass while the satellite quenching has an extra dependence on halo mass. The two best-fit models describe the red galaxy clustering and lensing equally well, but halo quenching provides significantly better fits to the blue galaxies above $$10^{11}h^{-2}M_{\odot}$$. The halo quenching model also correctly predicts the average halo mass of the red and blue centrals, showing excellent agreement with the direct weak lensing measurements of locally brightest galaxies. Models in which quenching is not tied to halo mass, including an age-matching model in which galaxy colour depends on halo age at fixed $$M_{*}$$, fail to reproduce the observed halo mass for massive blue centrals. We find similar critical halo masses responsible for the quenching of centrals and satellites  ($$\sim 1.5\times 10^{12}h^{-1}M_{\odot}$$), hinting at a uniform quenching mechanism for both, e.g., the virial shock-heating of infalling gas. The success of the iHOD halo quenching model provides strong evidence that the physical mechanism that quenches star formation in galaxies is tied principally to the masses of their dark matter halos rather than the properties of their stellar components.