A comprehensive model for evaluating the functionality of a multi-gear pump has been developed. The integrated model for assessing the functionality of a multi-gear pump contains a computational fluid dynamics analysis (CFD) model combined with a finite element method (FEM)-based strength model. Two submodels were linked: a CFD submodel to evaluate the internal pressure distribution of the pump and a structural FEM submodel to calculate the stresses and structural displacements of the pump due to fluid pressure. Finite element analysis in SolidWorks 2016 was used to evaluate the strength of the gear joints of the pump gears. As the pressure of the working fluid increases from 6 to 20 MPa, a linear increase in Mises stresses is observed. At the shaft, these stresses increase to 226.2 MPa, and at the tooth mouths, they reach a maximum value of 205.5 MPa. With the increase in torque on the drive shaft from 100 to 500 N·m, there is a significant increase in Mises stresses localized in the contact zones of the shaft with the drive gear. Analysis of the data obtained showed that the displacements caused by the pressure of the working fluid are insignificant compared to the displacements arising under the action of torque. With increasing pressure and torque, there is a tendency to decrease the safety factor, which indicates a decrease in the safety factor of the design of the multi-gear pump. The safety factor is not provided at a torque of 400–500 N·m. The simulation results are confirmed by correlation analysis. The average approximation error is 5–7%.