We report the detection of pulsed gamma-ray emission from the Geminga pulsar (PSR J0633+1746) between 15 GeV and 75 GeV. This is the first time a middle-aged pulsar has been detected up to these energies. Observations were carried out with the MAGIC telescopes between 2017 and 2019 using the low-energy threshold Sum-Trigger-II system. After quality selection cuts, ∼80 h of observational data were used for this analysis. To compare with the emission at lower energies below the sensitivity range of MAGIC, 11 years of Fermi-LAT data above 100 MeV were also analysed. From the two pulses per rotation seen by Fermi-LAT, only the second one, P2, is detected in the MAGIC energy range, with a significance of 6.3σ. The spectrum measured by MAGIC is well-represented by a simple power law of spectral index Γ = 5.62 ± 0.54, which smoothly extends the Fermi-LAT spectrum. A joint fit to MAGIC and Fermi-LAT data rules out the existence of a sub-exponential cut-off in the combined energy range at the 3.6σ significance level. The power-law tail emission detected by MAGIC is interpreted as the transition from curvature radiation to Inverse Compton Scattering of particles accelerated in the northern outer gap.