The invariant differential cross sections for inclusive π0 and η mesons at midrapidity were measured in pp collisions at s=2.76 TeV for transverse momenta 0.4 < pT< 40 GeV/c and 0.6 < pT< 20 GeV/c, respectively, using the ALICE detector. This large range in pT was achieved by combining various analysis techniques and different triggers involving the electromagnetic calorimeter (EMCal). In particular, a new single-cluster, shower-shape based method was developed for the identification of high-pT neutral pions, which exploits that the showers originating from their decay photons overlap in the EMCal. Above 4 GeV/c, the measured cross sections are found to exhibit a similar power-law behavior with an exponent of about 6.3. Next-to-leading-order perturbative QCD calculations differ from the measured cross sections by about 30% for the π0, and between 30–50% for the η meson, while generator-level simulations with PYTHIA 8.2 describe the data to better than 10–30%, except at pT< 1 GeV/c. The new data can therefore be used to further improve the theoretical description of π0 and η meson production.