Neutral pion and ηη meson invariant differential yields were measured in non-single diffractive p–Pb collisions at sNN−−−√sNN = 5.02 TeV with the ALICE experiment at the CERN LHC. The analysis combines results from three complementary photon measurements, utilizing the PHOS and EMCal calorimeters and the Photon Conversion Method. The invariant differential yields of π0π0 and ηη meson inclusive production are measured near mid-rapidity in a broad transverse momentum range oF0.3<pT<20 GeV/c0.3<pT<20 GeV/c and 0.7<pT<20 GeV/c0.7<pT<20 GeV/c, respectively. The measured η/π0η/π0 ratio increases with pTpT and saturates for pTpT > 4 GeV/cGeV/c at 0.483±0.015stat±0.015sys0.483±0.015stat±0.015sys. A deviation from mTmT scaling is observed for pT<pT< 2 GeV/cGeV/c. The measured η/π0η/π0 ratio is consistent with previous measurements from proton-nucleus and pp collisions over the full pTpT range. The measured η/π0η/π0 ratio at high pTpT also agrees within uncertainties with measurements from nucleus–nucleus collisions. The π0π0 and ηη yields in p–Pb relative to the scaled pp interpolated reference, RpPbRpPb, are presented for 0.3<pT<0.3<pT< 20 GeV/cGeV/cand 0.7<pT<0.7<pT< 20 GeV/cGeV/c, respectively. The results are compared with theoretical model calculations. The values of RpPbRpPb are consistent with unity for transverse momenta above 2 GeV/cGeV/c. These results support the interpretation that the suppressed yield of neutral mesons measured in Pb–Pb collisions at LHC energies is due to parton energy loss in the hot QCD medium.