In this paper, an efficient numerical method is proposed for the valuation of American better-of options based on the Black-Scholes model. Because of the existence of the optimal exercise boundary, the American better-of option satisfies a two dimensional parabolic linear complementarity problem on an unbounded domain. We first transform it into a one dimensional free boundary problem by a standard change of variables. And then the front-fixing transformation and the far field truncation are used to deal with the free boundary and the unbounded domain in succession, which yields a parabolic problem with unknown coefficient (free boundary) on a bounded regular domain. Furthermore, a finite element method is applied to discretize the resulting continuous system. The stability of the semi-discrete solution is also established. Meanwhile, Newton's method is used to solve the discretized system to obtain the free boundary and the option value simultaneously. The nonnegativity of the iteration solutions is also proved. Finally, numerical simulations are carried out to test the performance of the proposed method.