The colliding-wind binary system η Carinae has been identified as a source of high-energy (HE, below ∼100 GeV) and very high-energy (VHE, above ∼100 GeV) gamma rays in the last decade, making it unique among these systems. With its eccentric 5.5-year-long orbit, the periastron passage, during which the stars are separated by only 1–2 au, is an intriguing time interval to probe particle acceleration processes within the system. In this work, we report on an extensive VHE observation campaign that for the first time covers the full periastron passage carried out with the High Energy Stereoscopic System (H.E.S.S.) in its 5-telescope configuration with upgraded cameras. The VHE gamma-ray emission from η Carinae was detected during the periastron passage with a steep spectrum with the spectral index Γ = 3.3 ± 0.2stat ± 0.1syst. Together with previous and follow-up observations, we derive a long-term light curve sampling one full orbit, showing hints of an increase in the VHE flux towards periastron, but no hint of variability during the passage itself. An analysis of contemporaneous Fermi-LAT data shows that the VHE spectrum represents a smooth continuation of the HE spectrum. From modelling the combined spectrum, we conclude that the gamma-ray emission region is located at distances of ∼10–20 au from the centre of mass of the system and that protons are accelerated up to energies of at least several tera-electronvolts inside the system in this phase.