The spin dynamics of the spin-1/2 triangular-lattice antiferromagnet Cs₂CuBr₄ is probed by means of high-frequency electron paramagnetic resonance (EPR) spectroscopy. Temperature dependences of EPR parameters are studied in a broad temperature range between 1.4 and 200 K for different orientations of the applied magnetic field. In the high-temperature regime (T ≤ J/k_B), an unusually broad and anisotropic resonance line is detected, suggesting a sizeable Dzyaloshinskii-Moriya interaction. Employing the theory of exchange narrowing, the ratio of the Dzyaloshinskii-Moriya vector components, D_c/D_a ≈ 0.3, is estimated.