We present the discovery of the totally eclipsing long-period (P = 771.8 d) binary system WOCS 23009 in the old open cluster NGC 6819 that contains both an evolved star near central hydrogen exhaustion and a low-mass (0.45 Msun) star. This system was previously known to be a single-lined spectroscopic binary, but the discovery of an eclipse near apastron using data from the Kepler space telescope makes it clear that the system has an inclination that is very close to 90 degrees. Although the secondary star has not been identified in spectra, the mass of the primary star can be constrained using other eclipsing binaries in the cluster. The combination of total eclipses and a mass constraint for the primary star allows us to determine a reliable mass for the secondary star and radii for both stars, and to constrain the cluster age. Unlike well-measured stars of similar mass in field binaries, the low-mass secondary is not significantly inflated in radius compared to model predictions. The primary star characteristics, in combination with cluster photometry and masses from other cluster binaries, indicates a best age of 2.62+/-0.25 Gyr, although stellar model physics may introduce systematic uncertainties at the ~10% level. We find preliminary evidence that the asteroseismic predictions for red giant masses in this cluster are systematically too high by as much as 8%.