Gaussian elimination is basically the same as LU factorization. The routine xGETRF computes the LU factorization (e.g., DGETRF for real double precision matrices). The U factor corresponds to the matrix after Gaussian elimination. The U factor is stored in the upper triangular part (including the diagonal) of the matrix A on exit.
LU factorization / Gaussian elimination is commonly used to solve linear systems of equations. You can use the xGETRS routine to solve a linear system once you have computed the LU factorization.