Growing concern about how loss of biodiversity will affect ecosystems has stimulated numerous studies. Although most studies have assumed that species go extinct randomly, species often go extinct in order of their sensitivity to a stress that intensifies through time (such as climate change). Here we show that the consequences of random and ordered extinctions differ. Both depend on food-web interactions that create compensation; that is, the increase of some species when their competitors and/or predators decrease in density due to environmental stress. Compensation makes communities as a whole more resistant to stress by reducing changes in combined species densities. As extinctions progress, the potential for compensation is depleted, and communities become progressively less resistant. For ordered extinctions, however, this depletion is offset and communities retain their resistance, because the surviving species have greater average resistance to the stress. Despite extinctions being ordered, changes in the food web with successive extinctions make it difficult to predict which species will show compensation in the future. This unpredictability argues for 'whole-ecosystem' approaches to biodiversity conservation, as seemingly insignificant species may become important after other species go extinct.