Pair-density-wave (PDW) order is a superconducting state with a spatially modulated order parameter and is generally subleading to uniform superconductivity within the conventional weak-coupling BCS paradigm. Here, using a twisted bilayer checkerboard-lattice model, we show that momentum-space nonsymmorphic symmetry provides a generic mechanism that can instead promote PDW order to a leading weak-coupling instability. Combining a controlled Wilsonian renormalization-group analysis at charge neutrality with a finite-doping Bethe–Salpeter analysis, we determine the leading ordering tendencies among various competing instabilities. At charge neutrality, two symmetry-related quadratic band touchings support spin-singlet and spin-triplet PDW instabilities for different attractive interactions, whereas repulsive interactions favor quantum Hall insulators and finite-Q density-wave orders. Upon doping, the nonsymmorphic symmetry, together with time reversal, protects the Cooper logarithm in the finite-momentum Q pairing channel at generic fillings, allowing PDW order to remain a leading instability over the broad doping range studied. This mechanism extends naturally to moiré Chern bands: by breaking time-reversal symmetry while preserving the nonsymmorphic symmetry and inversion, we obtain a fully gapped topological spin-triplet PDW state with BdG Chern number C_BdG=8. Our results establish momentum-space nonsymmorphic symmetry as a general weak-coupling route to PDW superconductivity, including topological PDW states.