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Changing climate extremes and invasion by non-native species are two of the most prominent threats to native faunas. Predicting the relationships between global change and native faunas requires a quantitative toolkit that effectively links the timing and magnitude of extreme events to variation in species abundances. Here, we examine how discharge anomalies - unexpected floods and droughts - determine covariation in abundance of native and non-native fish species in a highly variable desert river in Arizona. We quantified stochastic variation in discharge using Fourier analyses on >15000 daily observations. We subsequently coupled maximum annual spectral anomalies with a 15-year time series of fish abundances (1994-2008), using Multivariate Autoregressive State-Space (MARSS) models. Abiotic drivers (discharge anomalies) were paramount in determining long-term fish abundances, whereas biotic drivers (species interactions) played only a secondary role. As predicted, anomalous droughts reduced the abundances of native species, while floods increased them. However, in contrast to previous studies, we observed that the non-native assemblage was surprisingly unresponsive to extreme events. Biological trait analyses showed that functional uniqueness was higher in native than in non-native fishes. We also found that discharge anomalies influenced diversity patterns at the meta-community level, with nestedness increasing after anomalous droughts due to the differential impairment of native species. Overall, our results advance the notion that discharge variation is key in determining community trajectories in the long term, predicting the persistence of native fauna even in the face of invasion. We suggest this variation, rather than biotic interactions, may commonly underlie covariation between native and non-native faunas, especially in highly variable environments. If droughts become increasingly severe due to climate change, and floods increasingly muted due to regulation, fish assemblages in desert rivers may become taxonomically and functionally impoverished and dominated by non-native taxa.