The density functional theory (DFT) was used to systematically investigate the effects of non-precious metal catalysts doped with different numbers of N atoms, graphene-CoNx (x = 1,2,3,4), on reaction mechanism of acetylene hydrochlorination reaction. It was found that three graphene-CoNx (x = 1,2,3) catalysts can activate C2H2 and HCl and absorb them on their surface. The electron transfer between the metal atom and the N and C atoms attached to it can be used as a descriptor of catalytic properties. Moreover, the catalytic activity of graphene-CoNx (x = 1,2,3) decreased with the decrease in the number of N atoms. One of the graphene-CoN3 catalysts has the lowest reaction energy barrier of 0.77 eV free energy. This work reveals the local coordination environment of N-doped graphene-loaded single-atom Co-catalyzed acetylene plus hydrogen chloride, which provides a theoretical basis for the design of efficient green catalysts.