The development of various machines and equipment containing parts or assemblies made of hyperelastic materials (e.g., rubber, silicone) is difficult because of the intricacies involved in the description of their mechanical properties. This is especially seen in calculations using simulation modeling. The behavior of hyperelastic materials is described by utilizing the results of research conducted with specialized equipment. This allows for the most accurate determination of their mechanical properties. Hyperelastic materials are widely used across diverse industries, encompassing mechanical engineering, the chemical and petrochemical sectors, cement production, and beyond. To determine the mechanical characteristics of SKR-788 silicone, batches of test samples were prepared, varying solely in the ratio of the base to the catalyst. Laboratory testing of silicone samples was performed on an Instron 4500 device, and data such as loads, displacements, and deformations were obtained. In order to verify the results of the tests against the results of simulation modeling in Ansys software, a model of the experimental sample was built. The obtained results of uniaxial tensile testing of the experimental sample were taken into account during the description of the material in the Mooney–Rivlin model. The calculation scheme for the test sample during simulation modeling is similar to the one used during its laboratory testing. Applying the load to the test sample during the simulation proceeded incrementally based on time. As a result of the work, the constants of the SKR-788 silicone material for the three-parameter Mooney–Rivlin model were defined (C10 = −0.10335 МРа, C01 = 0.57534 МРа, and C11 = 0.093309 МРа). As a result of simulation modeling of the experimental silicone sample, the values of its displacements and stresses were obtained. Upon comparing the stress values derived from the results of laboratory testing on the Instron 4500 equipment with those obtained from simulation modeling, a discrepancy of up to 7% was identified. For the first time, the characteristics of the SKR-788 silicone material have been established and verified. This will facilitate the design and research of various equipment and machine components made from hyperelastic materials.