During chronic viral infection, sustained antigen stimulation leads to exhaustion of virus-specific CD$8^+$ T cells, characterized by elevated expression of inhibitory receptors and progressive functional impairment, including loss of cytokine production, reduced cytotoxicity, and diminished proliferative capacity. In this paper, to investigate how T cell exhaustion influences viral persistence, we developed a within-host mathematical model integrating viral infection dynamics with adaptive immune responses. The model demonstrates three non-trivial equilibria: infection-free equilibrium $(S_1),$ uncontrolled-infection state $(S_2),$ and immune-controlled equilibrium $(S_3).$ Through dynamical systems analysis, we established the local stability of all states $(S_1-S_3)$ and prove global stability for both $S_1$ (complete viral clearance) and $S_2$ (chronic infection). Notably, the system exhibits Hopf bifurcations at $S_2$ and $S_3,$ with distinct critical thresholds governing oscillatory dynamics. Numerical simulations reveal that successful immune-mediated control of viral load and infected cell levels requires maintenance of low CD$8^+$ T cell exhaustion rates.