An American exchange option is a rainbow option with two underlying assets, whose pricing model is a two-dimensional free boundary problem and is equivalent to a parabolic variational inequality problem on a two-dimensional unbounded domain. The present work proposes an effective numerical method for this complex problem. We first reduce the problem into a one-dimensional linear complementarity problem (LCP) on a bounded domain based on a dimension reduction transformation, an a priori estimate for the optimal exercise boundary, and a far-field truncation technique. This LCP is then approximated by a finite element method with a geometric partition in the spatial direction and a backward Euler method with a uniform partition in the temporal direction. The convergence order of the fully discretized scheme is established as well. Further, according to the features of the discretized system, a primal-dual active-set (PDAS) method is imposed to solve this problem to obtain the option price and the optimal exercise boundary simultaneously. Finally, several numerical simulations are carried out to verify the theoretical results and effectiveness of the proposed method.