The process e+e-→ ηπγ is studied in the center-of-mass energy range 1.05–2.00 GeV using data with an integrated luminosity of 94.5 pb - 1 collected by the SND detector at the VEPP-2000 e+e- collider. The e+e-→ ηπγ cross section is measured for the first time. It is shown that the dominant mechanism of this reaction is the transition through the ωη intermediate state. The measured cross section of the subprocess e+e-→ ωη→ ηπγ is consistent with previous measurements in the e+e-→ π+π-πη mode. It is found, with a significance of 5.6 σ, that the process e+e-→ ηπγ is not completely described by hadronic vector-pseudoscalar intermediate states. The cross section of this missing contribution, which can originate from radiation processes, e. g. e+e-→ a(1450) γ, is measured. It is found to be 15–20 pb in the wide energy range from 1.3 to 1.9 GeV.