For compressible flow computations, the present paper extends the ETAU (enhanced time-accurate upwind) unstructured finite volume (FV) scheme to handle curved domain boundary with better accuracy. For the interior cells in the computational domain or the boundary cells with straight line boundary, the original ETAU scheme with second order accuracy in space and time is applied. For those boundary cells with the curved geometry, a more accurate Non-Uniform Rational B-Spline (NURBS) representation of the boundary is considered. The NURBS is commonly employed in computer aided design (CAD) to construct complex geometries. Here, it yields an exact geometry expression of complex boundary geometry. By combining ETAU with NURBS, the NURBS incorporated ETAU scheme (NETAU) is proposed for more accurate geometrical representation and fluxes evaluation. Details of the computing procedure of the geometry and surface fluxes for cells on the curved boundary, such as special transformation strategies and merging of ETAU and NURBS, are introduced and implemented. With NURBS, the NETAU scheme are geometrically versatile and more flexible. Several two-dimensional (2D) numerical cases are investigated to demonstrate the performance, computing efficiency and benefits of the NETAU scheme. The numerical results show that, for flows with low speed and high Reynolds number, the NETAU scheme provides more accurate pressure distribution on curved boundary than the original ETAU scheme. Meanwhile, the high-speed flow case shows that the NETAU scheme is still stable for high Mach number problem with shocks. Thus, the NETAU scheme potentially provides an accurate tool to describe complex geometry in computational fluid dynamics (CFD) simulations. It will help to reduce computational costs and enhances accuracy for flow domain dominated by complex geometries, with features such as high curvature and sharp edges.