A major challenge in performing reactions in biological systems is the requirement for low substrate concentrations, often in the micromolar range. We report that copper cross-linked single-chain nanoparticles (SCNPs) are able to significantly increase the efficiency of copper(I)-catalyzed alkyne-azide cycloaddition (CuAAC) reactions at low substrate concentration in aqueous buffer by promoting substrate binding. Using a fluorogenic click reaction and dye uptake experiments, a structure-activity study is performed with SCNPs of different size and copper content and substrates of varying charge and hydrophobicity. The high catalytic efficiency and selectivity are attributed to a mechanism that involves an enzyme-like substrate binding process. Saturation-transfer difference (STD) NMR spectroscopy, 2D-NOESY NMR, kinetic analyses with varying substrate concentrations, and computational simulations are consistent with a Michaelis-Menten, two-substrate, random-sequential enzyme-like kinetic profile. This general approach may prove useful for developing more-sustainable catalysts and agents for biomedicine and chemical biology.