A large literature shows that families with more resources are able to provide better learning environments and make more ambitious educational choices for their children. At the end of compulsory education, the result is a social-origin gap in school-track attendance and learning outcomes. Our paper analyses whether this gap further widens thereafter for children with comparable school achievement, and whether the gap varies by gender and migrant status. We examine graduation rates from higher education by combining a cohort study from Switzerland with a reweighting method to match students on their school track, grades, reading literacy and place of residence at the end of compulsory school. The one observed feature that sets them apart is their parents’ socio-economic status. When analysing their graduation rates 14 years later at the age of 30, we find a large social-origin gap. The rate of university completion at age 30 is 20 percentage points higher among students from the highest socio-economic status quartile than among students from the lowest quartile, even though their school abilities were comparable at age 16. This gap appears to be somewhat smaller among women than men, and among natives than migrants, but differences are not statistically significant. For men and women, migrants and natives alike, abundant parental resources strongly increase the likelihood of university graduation in Switzerland.