The Tarantula Nebula in the Large Magellanic Cloud is known for its high star formation activity. At its center lies the young massive star cluster R136, providing a significant amount of the energy that makes the nebula shine so brightly at many wavelengths. Recently, young massive star clusters have been suggested to also efficiently produce very high-energy cosmic rays, potentially beyond PeV energies. Here, we report the detection of very-high-energy γ-ray emission from the direction of R136 with the High Energy Stereoscopic System, achieved through a multicomponent, likelihood-based modeling of the data. This supports the hypothesis that R136 is indeed a very powerful cosmic-ray accelerator. Moreover, from the same analysis, we provide an updated measurement of the γ-ray emission from 30 Dor C, the only superbubble detected at TeV energies presently. The γ-ray luminosity above 0.5 TeV of both sources is (2–3) × 1035 erg s−1. This exceeds by more than a factor of 2 the luminosity of HESS J1646−458, which is associated with the most massive young star cluster in the Milky Way, Westerlund 1. Furthermore, the γ-ray emission from each source is extended with a significance of >3σ and a Gaussian width of about 30 pc. For 30 Dor C, a connection between the γ-ray emission and the nonthermal X-ray emission appears likely. Different interpretations of the γ-ray signal from R136 are discussed.