Based on a sample of 2.25×108J/ψ events taken with the BESIII detector at the BEPCII collider, we present the results of a study of the decay J/ψ→ηϕπ+π−. The Y(2175) resonance is observed in the invariant mass spectrum of ϕf0(980) with a statistical significance of greater than 10σ. The corresponding mass and width are determined to be M=2200±6(stat)±5(syst) MeV/c2 and Γ=104±15(stat)±15(syst) MeV, respectively, and the product branching fraction is measured to be B(J/ψ→ηY(2175),Y(2175)→ϕf0(980),f0(980)→π+π−)=(1.20±0.14(stat)±0.37(syst))×10−4. The results are consistent within errors with those of previous experiments. We also measure the branching fraction of J/ψ→ϕf1(1285) with f1(1285)→ηπ+π− and set upper limits on the branching fractions for J/ψ→ϕη(1405)/ϕX(1835)/ϕX(1870) with η(1405)/X(1835)/X(1870)→ηπ+π− at the 90% confidence level.