To investigate the effect of axial stress on the attenuation characteristics of stress wave peaks, stress wave propagation experiments with small disturbance of a sandstone bar were carried out by a modified split Hopkinson pressure bar test system. Then, effects of axial stress on the waveform, attenuation rate, temporal-spatial attenuation characteristics, and attenuation sensitivity factor of the peak were studied. The results showed that the presence or absence of axial stress has a significant effect on the waveform. With axial stress loading, both temporal and spatial attenuation rates undergo similar development stages, “nonlinear stage + linear stage,” in which the demarcation stress (σ/σc) is 30%. Under the same axial stress, the peak decreases exponentially with the propagation time and distance with different attenuation intensities. With increasing axial stress, the temporal and spatial response intensities also experience “nonlinear stage + linear stage.” However, the temporal and spatial attenuation coefficients undergo three stages, first a dramatic decrease, then gentle development, and finally a sharp increase, in which demarcation stresses (σ/σc) are 30% and 55%. The defined attenuation sensitivity factor can well describe the attenuation sensitivity of peaks to different axial stresses. The conclusions can provide a theoretical reference for rock mass stability analysis in blasting excavation.