Matching the number counts of high-z submillimetre-selected galaxies (SMGs) has been a long-standing problem for galaxy formation models. In this paper, we use 3D dust radiative transfer to model the submm emission from galaxies in the SIMBA cosmological hydrodynamic simulations, and compare predictions to the latest single-dish observational constraints on the abundance of 850 mu m-selected sources. We find good agreement with the shape of the integrated 850 mu m luminosity function, and the normalization is within 0.25 dex at >3 mJy, unprecedented for a fully cosmological hydrodynamic simulation, along with good agreement in the redshift distribution of bright SMGs. The agreement is driven primarily by SIMBA's good match to infrared measures of the star formation rate (SFR) function between z = 2 and 4 at high SFRs. Also important is the self-consistent on-the-fly dust model in SIMBA, which predicts, on average, higher dust masses (by up to a factor of 2.5) compared to using a fixed dust-to-metals ratio of 0.3. We construct a light-cone to investigate the effect of far-field blending, and find that 52 per cent of sources are blends of multiple components, which makes a small contribution to the normalization of the bright end of the number counts. We provide new fits to the 850 mu m luminosity as a function of SFR and dust mass. Our results demonstrate that solutions to the discrepancy between submm counts in simulations and observations, such as a top-heavy initial mass function, are unnecessary, and that submillimetre-bright phases are a natural consequence of massive galaxy evolution.