We calculate the axial couplings of mesons and baryons containing a heavy quark in the static limit using lattice QCD. These couplings determine the leading interactions in heavy hadron chiral perturbation theory and are central quantities in heavy quark physics, as they control strong decay widths and the light quark mass dependence of heavy hadron observables. Our analysis makes use of lattice data at six different pion masses, 227MeV<m π< 352MeV, two lattice spacings, a=0.085, 0.112fm, and a volume of (2.7fm)3. Our results for the axial couplings are g 1=0.449(51), g 2=0.84(20), and g 3=0.71(13), where g 1 governs the interaction between heavy-light mesons and pions and g 2,3 are similar couplings between heavy-light baryons and pions. Using our lattice result for g 3, and constraining 1/m Q corrections in the strong decay widths with experimental data for Σc(*) decays, we obtain Γ[Σb(*) →Λ bπ ±]=4.2(1.0),4.8(1.1), 7.3(1.6), 7.8(1.8)MeV for the Σb+, Σb-, Σb*+, Σb*- initial states, respectively. We also derive upper bounds on the widths of the Ξb′(*) baryons.