We consider the finite element approximation of a coupled fluid-structure interaction (FSI) sys- tem, which consists of a three-dimensional (3D) Stokes flow and a two-dimensional (2D) fourth- order Euler-Bernoulli or Kirchoff plate. The interaction of these parabolic and hyperbolic partial differential equations (PDE) occurs at the boundary interface. The vertical displacement of the plate dynamics evolves on the flat portion of the boundary, where the coupling conditions are implemented via the matching velocities of the plate and fluid flow, as well as the Dirichlet boundary trace of the pressure. This pressure term also acts as a coupling agent, as it appears as a forcing term on the flat, elastic plate domain. Our main focus in this work is to establish the well-posedness of the system using a Lagrange multiplier and to generate numerical results for the approximate solutions to the FSI model. For this, we discuss a numerical algorithm that sequentially solves the fluid and plate subsystems through an effective decoupling approach. Numerical results of test problems are presented to illustrate the performance of the proposed method.