During the main phase of evolution of a protoplanetary disk, accretion regulates the inner-disk properties, such as the temperature and mass distribution, and in turn, the physical conditions associated with planet formation. The driving mechanism behind accretion remains uncertain; however, one promising mechanism is the removal of a fraction of angular momentum via a magnetohydrodynamic (MHD) disk wind launched from the inner tens of astronomical units of the disk. This paper utilizes CO isotopologue emission to study the unique molecular outflow originating from the HD 163296 protoplanetary disk obtained with the Atacama Large Millimeter/submillimeter Array. HD~163296 is one of the most well-studied Class II disks and is proposed to host multiple gas-giant planets. We robustly detect the large-scale rotating outflow in the 12CO J=2-1 and the 13CO J=2-1 and J=1-0 transitions. We constrain the kinematics, the excitation temperature of the molecular gas, and the mass-loss rate. The high ratio of the rates of ejection to accretion (5 - 50), together with the rotation signatures of the flow, provides solid evidence for an MHD disk wind. We find that the angular momentum removal by the wind is sufficient to drive accretion through the inner region of the disk; therefore, accretion driven by turbulent viscosity is not required to explain HD~163296s accretion. The low temperature of the molecular wind and its overall kinematics suggest that the MHD disk wind could be perturbed and shocked by the previously observed high-velocity atomic jet. This paper is part of the MAPS special issue of the Astrophysical Journal Supplement.