Computational fluid dynamics (CFD) simulation is an important tool as it enables engineers to study different design options without a time-consuming experimental workload. However, the prediction accuracy of any CFD simulation depends upon the set boundary conditions and upon the applied rheological constitutive equation. In the present study the viscoelastic nature of an unfilled gum acrylonitrile butadiene rubber (NBR) is considered by applying the integral and time-dependent Kaye–Bernstein–Kearsley–Zapas (K-BKZ) rheological model. First, exhaustive testing is carried out in the linear viscoelastic (LVE) and non-LVE deformation range including small amplitude oscillatory shear (SAOS) as well as high pressure capillary rheometer (HPCR) tests. Next, three abrupt capillary dies and one tapered orifice die are modeled in Ansys POLYFLOW. The pressure prediction accuracy of the K-BKZ/Wagner model was found to be excellent and insensitive to the applied normal force in SAOS testing as well as to the relation of first and second normal stress differences, provided that damping parameters are fitted to steady-state rheological data. Moreover, the crucial importance of viscoelastic modeling is proven for rubber materials, as two generalized Newtonian fluid (GNF) flow models severely underestimate measured pressure data, especially in contraction flow-dominated geometries.