A Cu-based catalyst has the advantages of low price and high reserves as well as catalytic activity in acetylene hydrochlorination. In order to better design and prepare efficient Cu single-atom catalysts (SACs) for acetylene hydrochlorination, a single Cu atom anchored by different heteroatoms was investigated by constructing the CuX4 (X = B, C, N, O, P) structure using the spin-polarized density functional theory (DFT). According to the results, the N-doped substrate significantly increased the charge of the Cu atom, and B- and P-doped substrates provided multiple active sites, while the O-doped substrate could not stably anchor the single Cu atom. In general, different catalysts had different dominant mechanisms, and it was found that the charge transfer between single metal atoms and substrates could be used as a descriptor of the catalytic performance. This work reveals the effect mechanisms of the local coordination environment for Cu SACs catalyzing acetylene hydrochlorination, as well as provides theoretical insights for the rational design of efficient Cu-based catalysts.