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Cosmic rays are widely held to drive outflows from star-forming galaxies and profoundly influence galaxy evolution. Direct evidence for cosmic-ray carrying outflows is however lacking. At the same time there is increasing awareness of the importance of massive star clusters in the acceleration of cosmic rays in galaxies. Here we report on the discovery of a nascent outflow driven by the massive star cluster Westerlund 1. Giga-electronvolt gamma-ray emission coincident with a cavity visible in atomic hydrogen traces the emergence of a population of relativistic electrons out of the Galactic Disc. The emission is offset from tera-electronvolt gamma-ray radiation surrounding the cluster, but connects to it smoothly spectrally and spatially. The implied energy density of co-accelerated protons and nuclei, assuming standard non-thermal electron/proton injection efficiencies, is at least an order of magnitude higher than that in the general interstellar medium. These particles therefore have the potential to dynamically influence the outflow. This discovery suggests that cosmic-ray loaded outflows may be a common feature of young massive star clusters, with implications for the transport of cosmic rays into the halo of the Galaxy.