Elastomeric engine mounts are key components of automotive suspension and powertrain systems, playing a decisive role in vibration control, noise reduction, and ride comfort enhancement. Due to the nonlinear and viscoelastic behavior of elastomeric materials, accurately predicting the response of these components under various loading conditions requires advanced numerical analysis combined with experimental data. In this study, uniaxial tension-compression tests were first conducted on standard rubber specimens, followed by static and dynamic cyclic loading tests on the engine mount component. The elastomer behavior was subsequently simulated in Abaqus—independently of the mount component test—using a third-order Ogden hyperelastic model for static analysis and a 3-term Prony series viscoelastic model for dynamic analysis. Static analysis results revealed a simulated mount stiffness of 244 N/mm compared to an experimental value of 221 N/mm, indicating an error of 10.4%. In dynamic analysis, the effective simulated stiffness values at 1 Hz and 10 Hz (corresponding to rigid body vibration modes and road roughness frequencies) were calculated as 240 N/mm and 270 N/mm, respectively, whereas the corresponding experimental values were 256 N/mm (6.25% error) and 279 N/mm (3.2% error). The results demonstrate that the developed model possesses suitable accuracy in predicting both the static and dynamic stiffness of the component, although limitations were observed in capturing the frequency-stiffening trend under static preload. The findings of this research provide valuable quantitative data for the analysis and design of similar elastomeric engine mounts.