We report the observation of a frequency shift and splitting of the electron spin resonance (ESR) mode of the low-dimensional S=1/2 frustrated antiferromagnet Cs2CuCl4 in the spin-correlated state above the ordering temperature 0.62 K. The shift and splitting exhibit strong anisotropy with respect to the direction of the applied magnetic field and do not vanish in a zero field. The low-temperature evolution of the ESR is a result of the modification of the one-dimensional spinon continuum by the uniform Dzyaloshinskii-Moriya interaction within the spin chains.