We demonstrate, using a combination of experiment and density functional theory, that orbital ordering drives the formation of a one-dimensional (1D) S=1/2 antiferromagnetic spin chain in the 3D rocksalt structure of cesium superoxide (CsO2). The magnetic superoxide anion (O2(-)) exhibits degeneracy of its 2p-derived molecular orbitals, which is lifted by a structural distortion on cooling. A spin chain is then formed by zigzag ordering of the half-filled superoxide orbitals, promoting a superexchange pathway mediated by the p(z) orbitals of Cs(+) along only one crystal direction. This scenario is analogous to the 3d-orbital-driven spin chain found in the perovskite KCuF3 and is the first example of an inorganic quantum spin system with unpaired p electrons.