The statistical thermodynamics of binary mixtures of polyatomic species was developed based on a generalization in the spirit of the lattice-gas model and the quasi-chemical approximation (QCA). The new theoretical framework is obtained by combining: (i) the exact analytical expression for the partition function of non-interacting mixtures of linear k-mers and l-mers (species occupying k sites and l sites, respectively) adsorbed in one dimension, and its extension to higher dimensions; and (ii) a generalization of the classical QCA for multicomponent adsorbates and multisite-occupancy adsorption. This process is analyzed using the partial adsorption isotherms corresponding to both species of the mixture. Comparisons with analytical data from Bragg-Williams approximation (BWA) and Monte Carlo simulations are performed in order to test the validity of the theoretical model. Even though a good fitting is obtained from BWA, it is found that QCA provides a more accurate description of the phenomenon of adsorption of interacting polyatomic mixtures.