This paper is the first to analyze the robustness of goodness-of-fit for bivariate elliptical and archimedean copulas. To assess the tests robustness, we consider perturbations and outliers both in the dependence structure and the observations from the joint distribution. The Monte Carlo simulations show that independent of the underlying true copula, the GoF-test or chosen test statistic, even minor contaminations of the data can lead to a significant decrease in the GoF-tests power. In order to robustify the GoF-tests, several methods for the detection of multivariate outliers are applied to the contaminated data. The results show that the exclusion of outliers can have a beneficial effect on the power of the GoF-tests. Moreover, this robustification strategy improves the power of GoF-testing when used to identify the main component of a mixture copula. In the empirical risk management application, the practical usefullness of this strategy is exemplified for a set of bivariate portfolios.