We demonstrate a new Bayesian technique to invert color-magnitude diagrams of main-sequence and white dwarf stars to reveal the underlying cluster properties of age, distance, metallicity, and line-of-sight absorption, as well as individual stellar masses. The advantages our technique has over traditional analyses of color-magnitude diagrams are objectivity, precision, and explicit dependence on prior knowledge of cluster parameters. Within the confines of a given set of often-used models of stellar evolution, a single mapping of initial to final masses, and white dwarf cooling, and assuming photometric errors that one could reasonably achieve with the Hubble Space Telescope, our technique yields exceptional precision for even modest numbers of cluster stars. For clusters with 50-400 members and one to a few dozen white dwarfs, we find typical internal errors of σ([Fe/H]) ≤ 0.03 dex, σ(m - MV) ≤ 0.02 mag, and σ(AV) ≤ 0.01 mag. We derive cluster white dwarf ages with internal errors of typically only 10% for clusters with only three white dwarfs and almost always ≤5% with 10 white dwarfs. These exceptional precisions will allow us to test white dwarf cooling models and standard stellar evolution models through observations of white dwarfs in open and globular clusters.