A laser-driven electron-storage ring can produce nearly monochromatic, tunable X-rays in the keV energy regime by inverse Compton scattering. The small footprint, relative low cost and excellent beam quality provide the prospect for valuable preclinical use in radiography and tomography. The monochromaticity of the beam prevents beam hardening effects that are a serious problem in quantitative determination of absorption coefficients. These values are important e.g. for osteoporosis risk assessment. Here, we report quantitative computed tomography (CT) measurements using a laser-driven compact electron-storage ring X-ray source. The experimental results obtained for quantitative CT measurements on mass absorption coefficients in a phantom sample are compared to results from a rotating anode X-ray tube generator at various peak voltages. The findings confirm that a laser-driven electron-storage ring X-ray source can indeed yield much higher CT image quality, particularly if quantitative aspects of computed tomographic imaging are considered.