Neutral pion and eta meson invariant differential yields were measured in non-single diffractive p–Pb collisions at sqrt{s_{mathrm{NN}}} = 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 pi ^{0} and eta meson inclusive production are measured near mid-rapidity in a broad transverse momentum range of 0.3< p_{mathrm{T}} < 20 hbox { GeV}/c and 0.7< p_{mathrm{T}} < 20hbox { GeV}/c, respectively. The measured eta /pi ^{0} ratio increases with p_{mathrm{T}} and saturates for p_{mathrm{T}} > 4 hbox {GeV}/c at 0.483 pm 0.015_{mathrm{stat}} pm 0.015_{mathrm{sys}}. A deviation from m_{mathrm{T}} scaling is observed for p_{mathrm{T}}< 2 hbox {GeV}/c. The measured eta /pi ^{0} ratio is consistent with previous measurements from proton-nucleus and pp collisions over the full p_{mathrm{T}} range. The measured eta /pi ^{0} ratio at high p_{mathrm{T}} also agrees within uncertainties with measurements from nucleus–nucleus collisions. The pi ^{0} and eta yields in p–Pb relative to the scaled pp interpolated reference, R_{{mathrm{pPb}}}, are presented for 0.3 < p_{mathrm{T}}< 20 hbox {GeV}/c and 0.7 < p_{mathrm{T}}< 20 hbox {GeV}/c, respectively. The results are compared with theoretical model calculations. The values of R_{{mathrm{pPb}}} are consistent with unity for transverse momenta above 2 hbox {GeV}/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.