This article describes an original method for calculating stiffness and strength of multilayer beams (MBs) when the beam is deformed within elasticity limits. It draws on previous studies by the same author, but for the first time fully describes the proposed method, both the theory and its practical application. This method is very useful for engineering calculations and scientific research. The equations obtained allow establishing the positions of neutral layers and of the geometric and stiffness centers when the cross section of the beam is not symmetric to both x- and y-axes. This study also presents the equations for establishing the stiffness for bending and shear, and for calculating normal and shear stresses at any point of the beam’s cross section. The efficiency and practical application of this new method are demonstrated by analyzing how MB stiffness and strength change depending on the influence of certain factors, such as mechanical properties of layers, number of layers, and their arrangement and dimensions.