The vibration control of a sandwich beam with supported mass subjected to random support motion excitations can be performed using magnetorheological visco-elastomer core with adjustable dynamic properties. The periodic distributions of geometrical and physical parameters of the sandwich beam can improve its vibration response characteristics. To further improve characteristics or reduce responses, the quasi-periodic sandwich beam with supported mass under random excitations is studied. The facial layer thickness and core layer modulus of the sandwich beam are considered as quasi-periodic distributions. The partial differential equations for the horizontal and vertical coupling motions of the sandwich beam are derived and converted into ordinary differential equations for multi-degrees-of-freedom (DOFs) vibration. The expressions of frequency response and response spectral densities of the sandwich beam are obtained. Numerical results are given to illustrate the greatly improvable vibration response characteristics of the sandwich beam and the outstanding relative reduction localization of antiresonant responses. The proposed quasi-periodic distribution and analysis method can be used for the vibration control design of sandwich beams subjected to random excitations.