Success of alien plants is often attributed to high competitive ability. However, not all aliens become dominant, and not all natives are vulnerable to competitive exclusion. Here, we quantified competitive outcomes and its determinants, using response-surface experiments, in 48 pairs of native and naturalized alien annuals that are common or rare in Germany. Overall, aliens were not more competitive than natives. However, common aliens (invasive) were, despite strong limitation by intraspecific competition, more competitive than rare natives. This is because alien species had higher intrinsic growth rates than natives, and common species had higher intrinsic growth rates than rare ones. Strength of Interspecific competition was not related to status or commonness. Our work highlights the importance of including commonness in understanding invasion success. It suggests that variation among species in intrinsic growth rates is more important in competitive outcomes than inter- or intraspecific competition, and thus contributes to invasion success and rarity.