Recently, it was proposed that the orbital ordering of ${\ensuremath{\pi}}_{x,y}^{*}$ molecular orbitals in the superoxide ${\mathrm{CsO}}_{2}$ compound leads to the formation of spin-1/2 chains below the structural phase transition occurring at ${T}_{\mathrm{s}1}=61 \mathrm{K}$ on cooling. Here we report a detailed $X$-band electron paramagnetic resonance (EPR) study of this phase in ${\mathrm{CsO}}_{2}$ powder. The EPR signal appears as a broad line below ${T}_{\mathrm{s}1}$, which is replaced by the antiferromagnetic resonance below the N\'eel temperature ${T}_{\mathrm{N}}=8.3 \mathrm{K}$. The temperature dependence of the EPR linewidth between ${T}_{\mathrm{s}1}$ and ${T}_{\mathrm{N}}$ agrees with the predictions for the one-dimensional Heisenberg antiferromagnetic chain of $S=1/2$ spins in the presence of symmetric anisotropic exchange interaction. Complementary analysis of the EPR line shape, linewidth, and the signal intensity within the Tomonaga-Luttinger liquid (TLL) framework allows for a determination of the TLL exponent $K=0.48$. Present EPR data thus fully comply with the quantum antiferromagnetic state of spin-1/2 chains in the orbitally ordered phase of ${\mathrm{CsO}}_{2}$, which is therefore a unique $p$-orbital system where such a state could be studied.