Stellar clusters form by gravitational collapse of turbulent molecular clouds, with up to several thousand stars per cluster1. They are thought to be the birthplace of most stars and therefore play an important role in our understanding of star formation, a fundamental problem in astrophysics2,3. The initial conditions of the molecular cloud establish its dynamical history until the stellar cluster is born. However, the evolution of the cloud's angular momentum during cluster formation is not well understood4. Current observations have suggested that turbulence scrambles the angular momentum of the cluster-forming cloud, preventing spin alignment among stars within a cluster5. Here we use asteroseismology6-8 to measure the inclination angles of spin axes in 48 stars from the two old open clusters NGC 6791 and NGC 6819. The stars within each cluster show strong alignment. Three-dimensional hydrodynamical simulations of proto-cluster formation show that at least 50% of the initial proto-cluster kinetic energy has to be rotational in order to obtain strong stellar-spin alignment within a cluster. Our result indicates that the global angular momentum of the cluster-forming clouds was efficiently transferred to each star and that its imprint has survived several gigayears since the clusters formed.