We propose and demonstrate, first on simulated spectra and then experimentally, a novel approach to correct the undesired background distortions in the Brillouin spectra caused by molecular filter's absorption, fluorescent emission, ambient room light or any other constant contaminant. The developed multi-wavelength excitation Brillouin spectroscopy method computationally reconstructs the pure Brillouin component of the signal from multiple Brillouin spectra acquired using different excitation wavelengths. By removing the baseline distortions, the approach improves the goodness of fit of the Brillouin peaks, enabling accurate Brillouin shift and linewidth measurements from a wide range of challenging samples. In the present report, we explain the principle behind the method on a set of simulated spectra and present experimental application on an intentionally strongly-distorted spectrum. Utilizing the multi-excitation Brillouin spectroscopy approach, we successfully reconstruct Brillouin spectra of a highly-scattering sample, initially rendered not analyzable by excessive iodine absorption and contamination by out-of-focus light.