Using 1.31×109 J/ψ events collected by the BESIII detector at the BEPCII e+e- collider, we report the first observation of the h1(1380) in J/ψ→η′h1(1380) with a significance of more than ten standard deviations. The mass and width of the possible axial-vector strangeonium candidate h1(1380) are measured to be M=(1423.2±2.1±7.3) MeV/c2 and Γ=(90.3±9.8±17.5) MeV. The product branching fractions, assuming no interference, are determined to be B(J/ψ→η′h1(1380))×B(h1(1380)→K∗(892)+K-+c.c.)=(1.51±0.09±0.21)×10-4 in η′K+K-π0 mode and B(J/ψ→η′h1(1380))×B(h1(1380)→K∗(892)K̄+c.c.)=(2.16±0.12±0.29)×10-4 in η′KS0K±π mode. The first uncertainties are statistical and the second are systematic. Isospin symmetry violation is observed in the decays h1(1380)→K∗(892)+K-+c.c. and h1(1380)→K∗(892)0K̄0+c.c.. Based on the measured h1(1380) mass, the mixing angle between the states h1(1170) and h1(1380) is determined to be (35.9±2.6)°, consistent with theoretical expectations.