The minimum irradiance needed to overcome amplified spontaneous emission (ASE) of a seed beam injected into a laser amplifier is evaluated. The treatment is particularly applicable to extreme ultraviolet (EUV) and X-ray laser schemes to inject laser harmonic radiation as a seed into (1) plasma laser amplifiers and (2) free-electron lasers. Simple expressions and calculations are given for the minimum injected irradiance required for amplification of the injected seed beam to exceed ASE from the amplifier, including the effects of gain saturation, assuming one dimensional radiative transfer.