We present here a study of the magnetic properties of the antiferromagnetic dimer material CuVOF$_4$(H$_2$O)$_6cdot$H$_2$O, in which the dimer unit is composed of two different $S = 1/2$ species, Cu(II) and V(IV). An applied magnetic field of $mu_0H_{rm c1} = 13.1(1)~rm T$ is found to close the singlet-triplet energy gap, the magnitude of which is governed by the antiferromagnetic intradimer, $J_0 approx 21~rm K$, and interdimer, $J approx 1~rm K$, exchange energies, determined from magnetometry and electron-spin resonance measurements. The results of density functional theory (DFT) calculations are consistent with the experimental results and predicts antiferromagnetic coupling along all nearest-neighbor bonds, with the magnetic ground state comprising spins of different species aligning antiparallel to one another, while spins of the same species are aligned parallel. The magnetism in this system cannot be accurately described by the overlap between localized V orbitals and magnetic Cu orbitals lying in the Jahn-Teller (JT) plane, with a tight-binding model based on such a set of orbitals incorrectly predicting that interdimer exchange should be dominant. DFT calculations indicate significant spin density on the bridging oxide, suggesting instead an unusual mechanism in which intradimer exchange is mediated through the O atom on the Cu(II) JT axis.