Removing effects of sample heterogeneity through single-molecule and single-particle techniques has advanced many fields. While background free luminescence and scattering spectroscopy is widely used, recording the absorption spectrum only is rather difficult. Here we present an approach capable of recording pure absorption spectra of individual nanostructures. We demonstrate the implementation of single-particle absorption spectroscopy on strongly scattering plasmonic nanoparticles by combining photothermal microscopy with a supercontinuum laser and an innovative calibration procedure that accounts for chromatic aberrations and wavelength-dependent excitation powers. Comparison of the absorption spectra to the scattering spectra of the same individual gold nanoparticles reveals the blueshift of the absorption spectra, as predicted by Mie theory but previously not detectable in extinction measurements that measure the sum of absorption and scattering. By covering a wavelength range of 300 nm, we are furthermore able to record absorption spectra of single gold nanorods with different aspect ratios. We find that the spectral shift between absorption and scattering for the longitudinal plasmon resonance decreases as a function of nanorod aspect ratio, which is in agreement with simulations.