We present experimental constraints on the spin-dependent WIMP (weakly interacting massive particle)-nucleon elastic cross sections from LUX data acquired in 2013. LUX is a dual-phase xenon time projection chamber operating at the Sanford Underground Research Facility (Lead, South Dakota), which is designed to observe the recoil signature of galactic WIMPs scattering from xenon nuclei. A profile likelihood ratio analysis of 1.4×104 kg day of fiducial exposure allows 90% C.L. upper limits to be set on the WIMP-neutron (WIMP-proton) cross section of σn=9.4×10−41 cm2 (σp=2.9×10−39 cm2) at 33 GeV/c2. The spin-dependent WIMP-neutron limit is the most sensitive constraint to date.