Asexuality has major theoretical advantages over sexual reproduction. An important evolutionary puzzle, therefore, is why exclusively asexual metazoan lineages rarely endure. The Red Queen hypothesis posits that asexuality is rapidly extinguished by relentlessly coevolving parasites and pathogens. If so, any long-lasting asexual lineage must have unusual alternative mechanisms to deal with these biotic enemies. Bdelloid rotifers are freshwater invertebrates that abandoned sexual reproduction millions of years ago. Here, we show that cultured populations of bdelloids can rid themselves of a deadly fungal parasite through complete desiccation (anhydrobiosis) and disperse by wind to establish new populations in its absence. In Red Queen models, spatiotemporal escape can decouple and protect asexuals from coevolving enemies. Thus, our results may help to explain the persistence of the anciently asexual Bdelloidea.