This paper proposes the use of outlier detection methods from robust statistics and copula goodness-of-fit tests to identify components of mixture copulas. We first consider simulated data samples in which the true dependence structure is given by a mixture of two parametric copulas: one copula that is presumed to represent the true dependence structure and one disturbing copula. The Monte Carlo simulations show that the goodness-of-fit tests we consider significantly lose power when applied to mixtures of copulas with different tail dependencies. Several goodness-of-fit tests are shown to hold their nominal level when multivariate outliers are excluded, although this improvement comes at the price of a further loss in the tests' power. The usefulness of excluding outliers in copula goodness-of-fit testing is exemplified in an empirical risk management application.