Miscible displacement in porous media is a complex phenomenon governed by coupled nonlinear partial differential equations, often challenging to solve due to sharp concentration fronts, high Peclet numbers, and strong coupling between advection and diffusion processes. Traditional numerical methods face difficulties in capturing these features without introducing excessive numerical diffusion or oscillations. This study presents a novel approach combining the Semi-Lagrangian (SL) method with Isogeometric Analysis (IGA) to address these challenges effectively. The SL method mitigates the Courant-Friedrichs-Lewy (CFL) condition limitations and accurately handles the advection-dominated regime by tracing fluid particles along characteristic paths. Meanwhile, IGA's high-order continuity and geometric flexibility provide precise spatial discretization, enabling accurate representation of both the porous medium's geometry and the solution's gradients. Our proposed approach incorporates advanced interpolation schemes to ensure mass conservation and leverages IGA’s local refinement capabilities to resolve sharp interfaces. Numerical experiments demonstrate that this combined methodology significantly reduces numerical dispersion and achieves high accuracy in capturing complex flow and transport dynamics. This hybrid method holds promise for improving the fidelity of simulations in subsurface flow problems, offering insights into enhanced oil recovery, contaminant transport, and groundwater remediation.