We report momentum-resolved charge excitations in a one-dimensional (1D) Mott insulator studied using high resolution inelastic x-ray scattering over the entire Brillouin zone for the first time. Excitations at the insulating gap edge are found to be highly dispersive (momentum dependent) compared to excitations observed in two-dimensional Mott insulators. The observed dispersion in 1D cuprates ( SrCuO2 and Sr2CuO3) is consistent with charge excitations involving holons which is unique to spin-1/2 quantum chain systems. These results point to the potential utility of momentum-resolved inelastic x-ray scattering in providing valuable information about electronic structure of strongly correlated insulators.