An efficient adsorbent, CNTs–PAMAM–Ag, was prepared by grafting fourth-generation aromatic poly(amidoamine) (PAMAM) to carbon nanotubes (CNTs) and successive deposition of Ag nanoparticles. The FT–IR, XRD, TEM and XPS results confirmed the successful grafting of PAMAM onto CNTs and deposition of Ag nanoparticles. The absorption efficiency of CNTs–PAMAM–Ag was evaluated by estimating the adsorption of two toxic contaminants in water, viz., Pb(II) and As(III). Using CNTs–PAMAM–Ag, about 99 and 76% of Pb(II) and As(III) adsorption, respectively, were attained within 15 min. The controlling mechanisms for Pb(II) and As(III) adsorption dynamics were revealed by applying pseudo-first and second-order kinetic models. The pseudo-second-order kinetic model followed the adsorption of Pb(II) and As(III). Therefore, the incidence of chemisorption through sharing or exchanging electrons between Pb(II) or As(III) ions and CNTs–PAMAM–Ag could be the rate-controlling step in the adsorption process. Further, the Weber–Morris intraparticle pore diffusion model was employed to find the reaction pathways and the rate-controlling step in the adsorption. It revealed that intraparticle diffusion was not a rate-controlling step in the adsorption of Pb(II) and As(III); instead, it was controlled by both intraparticle diffusion and the boundary layer effect. The adsorption equilibrium was evaluated using the Langmuir, Freundlich, and Temkin isotherm models. The kinetic data of Pb(II) and As(III) adsorption was adequately fitted to the Langmuir isotherm model compared to the Freundlich and Temkin models.