Dendrimers are attractive macromolecules for a broad range of applications owing to their well-defined shapes and dimensions, highly branched and globular architectures, and opportunities for exploiting multivalency. Triazine dendrimers in particular offer advantages such as ease of synthesis, stability, well-defined spherical structure, multivalency, potential to achieve acceptable drug loadings, and low polydispersity. In this study, the potential utility of alkyne-azide “click” cycloadditions of first-, second-, and third-generation triazine dendrimers containing three or six alkynyl groups with benzyl azide was examined using copper catalysts. “Click-and-grow” and “grow-then-click” strategies were employed. For the first- and second- generation dendrimers, the desired triazole derivatives were obtained in high yields and purified by simple reprecipitation without column chromatography; however, some difficulties were observed in the preparation of third-generation dendrimers. The desired reaction proceeded under microwave irradiation as well as with simple heating. This click chemistry can be utilized for various melamine dendrimers that are fabricated with other amine linkers.