The production yield of the Λ(1520) baryon resonance is measured at midrapidity in Pb-Pb collisions at sNN = 2.76 TeV with the ALICE detector at the Large Hadron Collider (LHC). The measurement is performed in the Λ(1520)→pK- (and charge conjugate) hadronic decay channel as a function of the transverse momentum (pT) and collision centrality. The ratio of the pT-integrated production of Λ(1520) baryons relative to Λ baryons in central collisions is suppressed by about a factor of 2 with respect to peripheral collisions. This is the first observation of the suppression of a baryonic resonance at the LHC and the first 3σ evidence of Λ(1520) suppression within a single collision system. The measured Λ(1520)/Λ ratio in central collisions is smaller than the value predicted by the statistical hadronization model calculations. The shape of the measured pT distribution and the centrality dependence of the suppression are reproduced by the EPOS3 Monte Carlo event generator. The measurement adds further support to the formation of a dense hadronic phase in the final stages of the evolution of the fireball created in heavy-ion collisions, lasting long enough to cause a significant reduction in the observable yield of short-lived resonances.