Copper-catalyzed azide-alkyne cycloaddition (CuAAC) is one of the most versatile reactions in the “click chemistry” toolbox, and its development has made the synthesis of 1,4-triazole derivatives robust and efficient. In this work, we present a density functional theory (DFT) study on the mechanism of the CuAAC reaction catalyzed by a dicopper complex supported by a nonsymmetric 1,8-naphtyridine ligand bearing two different metal-coordinating substituents (i.e., −P(tBu)2 and −C(Me)(Py)2). The calculations showed that the cycloaddition of the azide to the alkyne occurs in a single concerted step, in contrast with the two-step mechanism proposed in the literature. The energies predicted for this step indicated that the 1,4-triazole isomer of the product is formed in a selective manner, in agreement with experiments. Further, the DFT results showed that there is a subtle and complex interplay between several variables, including the relative orientation of the two substrates, the position of the counter-anion, and the partial decoordination of the 1,8-naphtyridine ligand. A series of 90 transition state calculations showed that, on average, the impact of these variables is strong on the structures but soft on the energy barriers, highlighting the flexible nature of the bonding within the coordination sphere of the bimetallic core of the catalyst. The insight provided by this study will be valuable for the further development of dicopper catalysts for the CuAAC reaction.