In this work we examine the low-temperature magnetic properties of the two double-perovskite compounds Sr2CuWO6 and Sr2CuMoO6 using magnetic susceptibility, muon spin rotation and relaxation, and neutron powder diffraction measurements. Additionally, the most relevant spin exchange interaction constants are derived from ab initio electronic structure calculations, aided by x-ray absorption spectroscopy. The compounds exhibit quasi-two-dimensional magnetic properties, with broad maxima at Tmax = 83 and 95 K for Sr2CuWO6 and Sr2CuMoO6, respectively. However, three-dimensional long-range order takes place below TN = 24(1) and 28(2) K for Sr2CuWO6 and Sr2CuMoO6, respectively. Our results show that the low-dimensional magnetic correlations are mainly due to the significant next-nearest-neighbor interactions in the ab plane of the double-perovskite structure, whereas three-dimensional long-range magnetic order is caused by weaker next-nearest-neighbor interactions along the c axis. Next-nearest-neighbor interactions are also slightly frustrated by weaker nearest-neighbor interactions within the ab plane. Based on these results we predict the low-temperature magnetic structure in these compounds to be type-II antiferromagnetic order of the double-perovskite lattice.