The max-stable Husler-Reiss distribution which arises as the limit distribution of maxima of bivariate Gaussian triangular arrays has been shown to be useful in various extreme value models. For such triangular arrays, this paper establishes higher-order asymptotic expansions of the joint distribution of maxima under refined Husler-Reiss conditions. In particular, the rate of convergence of normalized maxima to the Husler-Reiss distribution is explicitly calculated. Our findings are supported by the results of a numerical analysis.