A highly dispersible and magnetically recoverable Cu-PEI@Fe$_{3}$O$_{4}$ MNPs catalyst was prepared and successfully applied in one-pot three-component coupling of terminal alkynes, sodium azide, and alkyl bromides/chlorides in water to give 1,4-disubstituted 1,2,3-triazoles with good to excellent yields. The catalyst was fully characterized with FT-IR, TGA, TEM, SEM, VSM, EDX, cyclic voltammetry, and ICP-AES spectroscopic techniques. Furthermore, the catalyst was easily recycled by an external magnet and successfully reused six times in the reaction without significant loss of its catalytic activity and copper leaching. The large-scale reaction was also carried out in the absence of any base and reducing agent even with 0.1 mol% of the catalyst in aqueous media, making this protocol a good candidate for practical applications.