H.E.S.S. observations of the 2021 periastron passage of PSR B1259-63/LS 2883

is a gamma-ray binary system that hosts a pulsar in an eccentric orbit, with a 3.4 year period, around an O9.5Ve star (LS 2883). At orbital phases close to periastron passages, the system radiates bright and variable non-thermal emission, for which the temporal and spectral properties of this emission are, for now, poorly understood. In this regard, very high-energy (VHE) emission is especially useful to study and constrain radiation processes and particle acceleration in the system. We report on an extensive VHE observation campaign conducted with the High Energy Stereoscopic System, comprised of approximately 100 hours of data taken over five months, from p -24$ days to $t_ p +127$ days around the system's 2021 periastron passage (where $t_ p $ is the time of periastron). We also present the timing and spectral analyses of the source. The VHE light curve in 2021 is consistent overall with the stacked light curve of all previous observations. Within the light curve, we report a VHE maximum at times coincident with the third X-ray peak first detected in the 2021 X-ray light curve. In the light curve -- although sparsely sampled in this time period -- we see no VHE enhancement during the second disc crossing. In addition, we see no correspondence to the 2021 GeV flare in the VHE light curve. The VHE spectrum obtained from the analysis of the 2021 dataset is best described by a power law of spectral index $ stat sys $, a value consistent with the spectral index obtained from the analysis of data collected with H.E.S.S. during the previous observations of the source. We report spectral variability with a difference of $ stat sys $ at $95 $ confidence intervals, between sub-periods of the 2021 dataset. We also detail our investigation into X-ray/TeV and GeV/TeV flux correlations in the 2021 periastron passage. We find a linear correlation between contemporaneous flux values of X-ray and TeV datasets, detected mainly after $t_ p +25$ days, suggesting a change in the available energy for non-thermal radiation processes. We detect no significant correlation between GeV and TeV flux points, within the uncertainties of the measurements, from $ t_p-23$ days to $ t_p+126$ days. This suggests that the GeV and TeV emission originate from different electron populations.