Paraxial transmission characteristic of a finite Olver-Gaussian beam (FOGB) in a LiNbO3 electro-optical crystal is investigated based on the generalized Huygens-Fresnel integral equation. The exact expression for the first-order FOGB passing through this electro-optical crystal orthogonal to the optical axis is derived. The contour graph of the FOGB intensity distribution on some transversal cross sections of the crystal, and the side view of this beam propagating evolution are discussed, respectively. The relation between external electric field and the central beam location, central relative beam intensity are also explored when wave propagating in the electro-optic crystal. Furthermore, the transmission characteristics of the FOGB in three kinds of traditional uniaxial crystal orthogonal to the optical axis are discussed as well.