We develop a methodology for extracting the Kohn-Sham angular momentum of excitons in realistic systems from time-dependent density functional theory. For small systems the exciton populations can be calculated analytically, which allows us to test the methodology for a three-arm H2 molecular ring and a pair of such rings. For larger systems the developed methodology opens a venue to determine the angular momentum of excitons by first principles calculations. A chain of twenty three-arm H2 molecular rings and a triphenylphosphine molecule are investigated as illustrative examples. It is demonstrated that the angular momentum is conserved during the absorption of twisted light.