The influence of the strength of intermolecular interactions on mass diffusive processes remains poorly understood for mixtures of associative liquids with dissolved gases. For contributing to a fundamental understanding of the interplay between liquid structures and mass diffusivities in such systems, dynamic light scattering, Raman spectroscopy, and molecular dynamics simulations were used in this work. As model systems, binary mixtures consisting of the gases hydrogen, helium, nitrogen, carbon monoxide, or carbon dioxide dissolved in ethanol, 1-hexanol, or 1-decanol were selected. Experiments and simulations were performed at macroscopic thermodynamic equilibrium close to infinite dilution of solute for temperatures between 303 and 423 K. The Fick diffusion coefficients and self-diffusivities of the gas solutes increase with increasing temperature, decreasing alkyl chain length of the 1-alcohols, and decreasing molar mass of the solutes except for helium and hydrogen showing the opposite behavior. The analysis of the liquid structure of the mixtures showed that the fraction of hydrogen-bonded alcohol molecules decreases with increasing alkyl chain length and temperature. From the obtained structure-property relationships, a new correlation was developed to predict mass diffusivities in binary mixtures consisting of n-alkanes or 1-alcohols with dissolved gases close to infinite dilution within 10% on average.