We report on high-field electron spin resonance (ESR) studies of magnetic excitations in the spin-1/2 triangular-lattice antiferromagnet Cs₂CuBr₄. Frequency-field diagrams of ESR excitations are measured for different orientations of magnetic fields up to 25 T. We show that the substantial zero-field energy gap, K, observed in the low-temperature excitation spectrum of Cs₂CuBr₄, (Zvyagin et al 2014 Phys. Rev. Lett.112 077206) is present well above T_N. Noticeably, the transition into the long-range magnetically ordered phase does not significantly affect the size of the gap, suggesting that even below T_N the high-energy spin dynamics in Cs₂CuBr₄ is determined by short-range-order spin correlations. The experimental data are compared with results of model spin-wave-theory calculations for spin-1/2 triangular-lattice antiferromagnet.