We present the first theoretical prediction of the partial decay width of the process J/ψ→γη1, where η1 is the lightest flavor singlet 1−+ hybrid meson. Our Nf=2 lattice QCD calculation at mπ≈350 MeV results in the η1 mass mη1=2.23(4) GeV and the related electromagnetic form factors M1(0)=−4.73(74) MeV, E2(0)=1.18(22) MeV, which give Γ(J/ψ→γη1)=2.04(61) eV. These form factors can be applied to the physical Nf=3 case, where there should be two hybrid mass eigenstates η1(l) and η1(h) due to the singlet-octet mixing. It is shown that the ratio of the branching fractions Br(J/ψ→γη1(l,h)→γηη′) is inversely proportional to the ratio of the total widths of η1(l,h). Given our results and the mixing angle derived by a previous lattice study, whether η1(1855) is assigned to be η1(1) or η1(h), the observed branching fraction J/ψ→η1(1855)→γηη′ implies a very large coupling of the octet η1 to ηη′. This should be investigated in future studies.