We present an EVI-space-time Galerkin method applied to dynamic analysis in fluid-saturated porous materials. The physical model is based on a materially incompressible solid skeleton saturated by a barotropic fluid. The deformation of the solid matrix is described by a compressible Neo-Hookean material law. The model equations are formulated in the Lagrangian description of the solid skeleton. In respect of the numerical modeling, by use of the Embedded Velocity Integration (EVI) technique, the governing set of second-order time-dependent equations is transformed to a first-order one, which is in turn solved by a time-discontinuous Galerkin method. In addition, a stability factor α, describing the embedded integration scheme of the displacement-velocity relation, is introduced. Depending on the chosen value of the α factor, the stability property of the overall solution can be enforced. Numerical experiments demonstrate the superior performance of the proposed method with respect to accuracy and low numerical damping in comparison with conventional time-stepping schemes.