<p>We propose a discrete-time Leslie–Gower predator–prey model incorporating prey refuge. By employing the piecewise constant argument method, we establish the biological feasibility and non-negativity of solutions. A local stability analysis of the positive equilibrium reveals that stronger refuge protection enhances prey density and may lead to population outbreaks. Using normal form theory and bifurcation analysis, we further investigate the onset of complex dynamical behaviors, including Neimark–Sacker and period-doubling bifurcations. A two-parameter study is conducted to characterize chaotic dynamics through Lyapunov exponents, power spectrum analysis, and the Kaplan–Yorke dimension. In addition, we introduce a novel chaos control strategy based on positivity-preserving multiplicative state feedback, which effectively stabilizes and suppresses chaotic oscillations without violating biological constraints. Numerical simulations are presented to confirm and illustrate the theoretical findings, demonstrating the robustness of the model.</p>