Rabies remains a critical zoonotic threat characterized by complex transmission dynamics between domestic dogs and human populations. Despite its severity, traditional integer-order models often overlook the hereditary properties and non-local memory effects inherent in epidemiological spread, necessitating more robust mathematical frameworks. To address this, this study presents a fractional-order mathematical model for simulating the nonlinear behaviors of rabies infection. Fractional calculus tools are utilized to better capture non-locality interactions and memory influences, establishing a more precise representation of epidemiological processes than conventional models. The dynamics are simulated through a system of eight fractional differential equations coupling canine and human populations, where the fractional-order formulation provides the flexibility to represent diverse disease progression scenarios. The analytical rigor of the model is established through investigations of existence, uniqueness, positivity, and boundedness of solutions. Furthermore, the stability characteristics and bifurcation scenarios of equilibrium points are explored, complemented by a sensitivity analysis that reveals the dependence of the basic reproduction number on key parameters. Our results demonstrate that increasing recovery or vaccination rates can stabilize the disease-free equilibrium. Finally, an optimal control scheme is employed to suppress the infection spread. These findings provide valuable insights into rabies control strategies, highlighting the advantages of fractional-order modeling in epidemiology and offering practical guidance for public health interventions which better help achieving Sustainable Development Goals (SDGs) related to worldwide good healthy lives.