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This research is devoted to semi-analytical stress and large deformation analyses of finite-length thick-walled incompressible Mooney–Rivlin hyperelastic cylindrical pressure vessels/pipes with fixed ends, employing the three-dimensional theory of hyperelasticity. The formulation accounts for the thickness reduction in both radial and axial directions. The governing system of equations and the boundary conditions are assembled using a large-weights penalty method, after utilizing central, forward, and backward second-order point collocation expressions for the discretization of the thick hollow cylinder in both longitudinal and radial directions. The resulting coupled highly nonlinear two-dimensional equations are solved by the Newton–Raphson iterative updating technique. The results include the radial, axial, and three-dimensional distributions of the deformations and all the stress components. Comprehensive parametric studies are accomplished to evaluate the effects of the hyperelasticity parameters, boundary conditions, and thickness and length-to-radius ratios of the pressure vessel/pipe. Results reveal that the sections located in the vicinity of the fixed edges and the mid-section of the cylinder undergo the highest stresses, at low and high pressures, respectively. Moreover, while the pipe undergoes an overall expansion due to the internal pressure, it experiences a bending moment that induces compressive stresses at the outer boundary, due to the fixed edge. The magnitudes of the radial deformation and the circumferential stress significantly grow by increasing the cylinder length and the nonlinearities in the cylinder responses and behaviors increase by increasing the cylinder length.