A two-dimensional electron gas (2DEG) can be formed at an interface between two insulators such as LaAlO₃ and SrTiO₃ without any additional doping. Nevertheless, structural imperfections or defects at the interface, which may arise during the growth process or be of another origin, can definitely affect 2DEG properties either increasing or reducing the 2DEG density. In this work, we study 2DEG formation at both perfect and imperfect interfaces of LaAlO₃/SrTiO₃ heterostructures in the framework of the density functional theory. Using a first-principles Green function method within a coherent potential approximation, we investigate consistently the development of the 2DEG density starting from a defectless interface and introducing sequentially various kinds of long-range disorder such as cation intermixing and oxygen vacancies. Finally, we evaluate the 2DEG carrier density from the Fermi surface cross sections and effective masses of the carriers, which may be directly related to the multiple transport phenomena in the system.