PSR B1259-63/LS 2883 is a binary system consisting of a 48 ms pulsar orbitting around a Be star with an orbital period of ~3.4 years. The system was detected at very high energies (VHE; E > 100 GeV) by the High Energy Stereoscopic System (H.E.S.S.) during its periastron passages in 2004 and 2007. Here we present new H.E.S.S. observations corresponding to its last periastron passage, which occurred on December 15th 2010. These new observations partially overlap with the beginning of a spectacular gamma-ray flare reported by the Fermi-LAT. The H.E.S.S. observations show both flux and spectral properties similar to those reported in previous periastron passages, without any signature of the emission enhancement seen at GeV energies. A careful statistical study based on the Fermi and H.E.S.S. lightcurves leads to the conclusion that the GeV and TeV emission during the flare have a different physical origin. This conlusion, in turn, allows to use Fermi-LAT measurements of the GeV flux as upper limits for the modeling of the VHE emission.