Anisotropic elliptic interface problems are important but hard to solve either analytically or numerically. There is limited literature on numerical methods based on structured meshes. Finite element methods are often used, but the usual average error estimates cannot guarantee accuracy of the solution near or at the interface. For finite difference methods, it is challenging to discretize mixed derivatives and carry out the convergence analysis. In this paper, a new finite element-finite difference (FE-FD) method that combines a finite element discretization (away from the interface) whose coefficient matrix is a symmetric semi-positive definite, with a finite difference discretization (near or on the interface) whose coefficient matrix part has properties of an M-matrix, is developed. An interpolation scheme based on the immersed interface method is also applied to compute the normal derivative of solution (or gradient) accurately from each side of the interface. Error analysis and numerical experiments are also presented.