We report on the amplification of an optical vortex beam carrying orbital angular momentum via induced narrow Raman gain in an ensemble of cold cesium atoms. A 20% single-pass Raman gain of a weak vortex signal field is observed with a spectral width of order of 1 MHz, much smaller than the natural width, demonstrating that the amplification process preserves the phase structure of the vortex beam. The gain is observed in the degenerated two-level system associated with the hyperfine transition 6S1/2(F = 3) ↔ 6P3/2(F' = 2) of cesium. Our experimental observations are explained with a simple theoretical model based on a three-level Λ system interacting coherently with the weak Laguerre-Gauss field and a strong coupling field, including an incoherent pumping rate between the two degenerate ground-states.