Neutral pion and η meson production in the transverse momentum range 1 <pT< 20 GeV/c have been measured at midrapidity by the ALICE experiment at the Large Hadron Collider (LHC) in central and semicentral Pb-Pb collisions at sNN = 2.76 TeV. These results were obtained using the photon conversion method as well as the Photon Spectrometer (PHOS) and Electromagnetic Calorimeter detectors. The results extend the upper pT reach of the previous ALICE π0 measurements from 12 to 20 GeV/c and present the first measurement of η meson production in heavy-ion collisions at the LHC. The η/π0 ratio is similar for the two centralities and reaches at high pT a plateau value of 0.457 ± 0.013stat ± 0.018syst. A suppression of similar magnitude for π0 and η meson production is observed in Pb-Pb collisions with respect to their production in pp collisions scaled by the number of binary nucleon-nucleon collisions. We discuss the results in terms of Next to Leading Order (NLO) pQCD predictions and hydrodynamic models. The measurements show a stronger suppression than observed at lower center-of-mass energies in the pT range 6 <pT< 10 GeV/c. For pT< 3 GeV/c, hadronization models describe the π0 results while for the η some tension is observed.