The scattering of dark matter (DM) particles with sub-GeV masses off nuclei is difficult to detect using liquid xenon-based DM search instruments because the energy transfer during nuclear recoils is smaller than the typical detector threshold. However, the tree-level DM-nucleus scattering diagram can be accompanied by simultaneous emission of a bremsstrahlung photon or a so-called “Migdal” electron. These provide an electron recoil component to the experimental signature at higher energies than the corresponding nuclear recoil. The presence of this signature allows liquid xenon detectors to use both the scintillation and the ionization signals in the analysis where the nuclear recoil signal would not be otherwise visible. We report constraints on spin-independent DM-nucleon scattering for DM particles with masses of 0.4–5  GeV/c2 using 1.4×104  kg day of search exposure from the 2013 data from the Large Underground Xenon (LUX) experiment for four different classes of mediators. This analysis extends the reach of liquid xenon-based DM search instruments to lower DM masses than has been achieved previously.