Massive star-forming galaxies at high redshift require a supply of molecular gas from their gas reservoirs that is replenished by infall from the surrounding circumgalactic medium to sustain their immense star formation rates. Our knowledge of the extent and morphology of cold-gas reservoirs of early galaxies is still in its infancy, however. We present the results of stacking more than 80 hours of JVLA observations of CO(1–0) emission, which traces the cold molecular gas, in 19 z = 2.0 ‑ 4.5 dusty star-forming galaxies from the AS2VLA survey. The visibility-plane stack reveals extended emission with a half-light radius of 3.8 ± 0.5 kpc, which is a factor of 2–3 more extended than the dust-obscured star formation and 1.4 ± 0.2× more extended than the stellar emission revealed by the JWST. Stacking the [C I](1–0) observations for 10 galaxies from our parent sample yielded a half-light radius ≤2.6 kpc, which is marginally smaller than CO(1–0). The CO(1–0) size is also comparable to that of the [C II] haloes detected around high-redshift star-forming galaxies. This suggests that these arise from molecular gas. Photo-dissociation region modelling indicates that the extended CO(1–0) emission arises from clumpy dense clouds and not from smooth diffuse gas. Our results show that the bulk (up to 80%) of the molecular gas in these galaxies resides outside the star-forming region with only a small part directly contributing to the star formation.