We present spatially resolved kinematics of 27 ALMA-identified dust-obscured star-forming galaxies (DSFGs) at z ∼ 1.3–2.6, as traced by Hα emission using VLT/KMOS near-infrared integral field spectroscopy from the “KMOS-ALMA Observations of Submillimetre Sources” (KAOSS) Large Programme. We derive Hα rotation curves and velocity dispersion profiles for the DSFGs, and find that among the 27 sources with bright, spatially extended Hα emission, 24 display evidence for disc-like kinematics. We measure a median inclination-corrected velocity at 2.2 Rd of vrot = 190 ± 40 km s−1 and intrinsic velocity dispersion of σ0 = 87 ± 6 km s−1 for these disc-like sources. The kinematics yield median circular velocities of vcirc = 230 ± 20 km s−1 and dynamical masses within 2Re (∼ 7 kpc radius) of Mdyn = (1.1 ± 0.2) × 1011 M⊙. Compared to less actively star-forming galaxies, KAOSS DSFGs are both faster rotating with higher intrinsic velocity dispersions, but have similar vrot/σ0 ratios, median v/σ0 = 2.5 ± 0.5. We suggest that the kinematics of the DSFGs are primarily rotation supported but with a non-negligible contribution from pressure support, which may be driven by star formation or mergers/interactions. We estimate the normalisation of the stellar mass Tully-Fisher relation (sTFR) for the disc-like DSFGs and compare it with local studies, finding no evolution at fixed slope between z ∼ 2 and z ∼ 0. Finally, we show that the kinematic properties of the DSFG population are consistent with them evolving into massive early-type galaxies, the dominant z ∼ 0 population at these masses.