We have investigated how a magnetic field applied perpendicular to the CuO2 planes of the near-optimally hole-doped high-temperature superconductor La1.855Sr0.145CuO4 (Tc ≈36 K) influences the low-energy magnetic excitation spectrum. Our detailed single-crystal neutron scattering experiments reveal that the gap to magnetic excitations falls off linearly with increasing field and reaches zero at the magnetic field µ0Hs=7±1 T required to induce long-range incommensurate magnetic order. A comparison with the electron-doped cuprate Nd1.85Ce0.15CuO4 is made and the possible link between field-induced magnetic order and Fermi surface reconstruction in cuprates is discussed.