We achieve control over the emission wavelength of a random laser experimentally by adjusting the amount of absorption of emission light. The random laser consists of elastic scatterers in a high quantum efficiency gain medium to which a nonfluorescent dye is added. This nonfluorescent dye shifts the net gain curve and thereby the emission peak of the random laser. Mode competition effects are observed and explained. We interpret our results using a theoretical model that describes the spectral response of the system as a function of the effective cavity decay time and (re)absorption processes.