Why do some alien species introductions fail while others are successful? One key hypothesis is that in order for a species to invade a new range, it must initially establish successfully in conditions within its native climatic niche (NCN). Yet, this hypothesis has largely remained untested at the level of individual release events. Using a dataset of 989 introductions of 177 mammal species worldwide, we show that climatic matching to the NCN, measured by a new Niche Margin Index (NMI), is a stronger predictor of establishment success than most previously tested life-history attributes and historical factors. Contrary to traditional climatic suitability metrics derived from species distribution models, the NMI is based on niche margins and provides a measure of how distant a site is inside or, importantly, outside the NCN. Integrating NMI as a measure of NCN-matching in risk assessments could improve efforts to prevent invasions and avoid costly eradications.