We propose a NURBS-based interface-tracking technique as a means of representing high-order interfaces. This technique is combined with the Enriched Finite Element Method (E-FEM) to simulate gas-liquid interfacial dynamics. While the E-FEM framework accurately captures strong and weak pressure at the interface on non-conforming meshes, the NURBS representation of the interface provides high geometric precision for normal and curvature computation. This is key to effectively suppressing the parasitic currents that arise due to erroneous local curvature calculations. NURBS-based interface representations dramatically improve the overall accuracy on significantly coarser meshes compared to conventional interface capturing techniques (e.g. the level-set method). The interface is represented by a closed, periodic NURBS curve, and its control points are advected using a fourth-order Runge–Kutta scheme. The corresponding velocities are calculated using a Galerkin projection, which couples the finite element method (FEM) velocity field with the NURBS parametric space. This yields a compact linear system. This Lagrangian interface-tracking strategy guarantees mass conservation and provides a high-order geometric representation of the interfaces, even when two interfaces are approaching each other. The proposed technique has been verified by solving benchmark problems, including the 2D Leveque vortex test, which demonstrates excellent shape recovery and area conservation. Furthermore, a 2D simulation of a bubble rising is presented to validate the framework combining E-FEM with NURBS-based interface-tracking in a physically driven multiphase flow scenario. The results demonstrate the method's ability to handle buoyancy-driven interface deformation with accurate, curvature-dependent surface tension forces. These results confirm that integrating the proposed NURBS-based interface tracking offers a robust, geometrically accurate alternative to Eulerian approaches, such as the conservative level-set method, for scenarios demanding high-curvature precision.