A linear eigenvalue problem governed by a second order elliptic equation with separate and general boundary conditions is considered and a new monotonicity result on the principal eigenvalue with respect to the coefficient of the advection term is established. The main approach is based on the functional proposed by Liu and Lou and a key finding lies in the nice properties of the associated Frechet operator when conned at suitable points and function spaces. As an application, this monotonicity result is used to study a class of competitive parabolic systems and the so-called “exclusion principle” is observed in a larger parameter region than several existing works, which is a nontrivial improvement.