This study analyzed the relative importance of different cognitive abilities for solving complex mathematical word problems (CWPs)—a demanding task of high relevance for diverse fields and contexts. We investigated the effects of spatial, verbal, numerical, and general reasoning abilities as well as gender on CWP performance among N = 1282 first-year university engineering students. Generalized linear mixed models unveiled significant unique effects of spatial ability, β = 0.284, verbal ability, β = 0.342, numerical ability, β = 0.164, general reasoning, β = 0.248, and an overall gender effect in favor of male students, β = 0.285. Analyses revealed negligible to small gender effects in verbal and general reasoning ability. Despite a gender effect in spatial ability, d = 0.48, and numerical ability, d = 0.30—both in favor of male students—further analyses showed that effects of all measured cognitive abilities on CWP solving were comparable for both women and men. Our results underpin that CWP solving requires a broad facet of cognitive abilities besides mere mathematical competencies. Since gender differences in CWP solving were not fully explained by differences in the four measured cognitive abilities, gender-specific attitudes, beliefs, and emotions could be considered possible affective moderators of CWP performance.