Modern drug delivery research focuses on developing biodegradable nanopolymer systems. The present study proposed a polymer-based composite nanogel as a transdermal drug delivery system for the pH-responsive targeted and controlled delivery of anticancer drug doxorubicin (DOX). Nanogels have properties of both hydrogels and nanomaterials. The β-cyclodextrin-based nanogels can enhance the loading capacity of poorly soluble drugs and promote a sustained drug release. The β-cyclodextrin-grafted methacrylic acid conjugated hyaluronic acid composite nanogel was successfully synthesized. β-Cyclodextrin was first grafted onto methacrylic acid. The composite nanogel-based drug carrier was prepared by controlled radical polymerization (CRP) of β-cyclodextrin-grafted methacrylic acid with hyaluronic acid. The doxorubicin-loaded carrier was characterized by Fourier transform infrared (FTIR) spectroscopy, nuclear magnetic resonance (NMR) spectroscopy, ultraviolet-visible (UV-vis) spectroscopy, zeta potential analysis, dynamic light scattering (DLS), scanning electron microscopy (SEM), and transmission electron microscopy (TEM). The drug loading and release efficiencies were carried out at different pH levels. The maximum drug loading and encapsulation efficiencies of the synthesized final nanogel composite material at pH 8.0 were 86.44 ± 2.12 and 96.07 ± 2.01%, respectively. The DOX-loaded final material showed a 90.0 ± 2.6% release percentage of DOX at pH 5.5, whereas at pH 7.4, the release percentage of DOX was observed to be only 35.0 ± 0.3%. In vitro swelling, degradation, hemocompatibility, drug release kinetics, cytotoxicity, apoptosis, cell colocalization, skin irritation, and skin permeation studies, along with in vivo pharmacokinetic studies, were performed to prove the efficacy of the synthesized nanogel composite as a transdermal carrier for doxorubicin.