MIGHTEE is a galaxy evolution survey using simultaneous radio continuum, spectro-polarimetry, and spectral line observations from the South African MeerKAT telescope. When complete, the survey will image ∼20 deg2 over the COSMOS, E-CDFS, ELAIS-S1, and XMM-LSS extragalactic deep fields with a central frequency of 1284 MHz. These were selected based on the extensive multiwavelength datasets from numerous existing and forthcoming observational campaigns. Here we describe and validate the data processing strategy for the total intensity continuum aspect of MIGHTEE, using a single deep pointing in COSMOS (1.6 deg2) and a three-pointing mosaic in XMM-LSS (3.5 deg2). The processing includes the correction of direction-dependent effects, and results in thermal noise levels below 2~μJy beam−1 in both fields, limited in the central regions by classical confusion at ∼8′′ angular resolution, and meeting the survey specifications. We also produce images at ∼5′′ resolution that are ∼3 times shallower. The resulting image products form the basis of the Early Science continuum data release for MIGHTEE. From these images we extract catalogues containing 9,896 and 20,274 radio components in COSMOS and XMM-LSS respectively. We also process a close-packed mosaic of 14 additional pointings in COSMOS and use these in conjunction with the Early Science pointing to investigate methods for primary beam correction of broadband radio images, an analysis that is of relevance to all full-band MeerKAT continuum observations, and wide field interferometric imaging in general. A public release of the MIGHTEE Early Science continuum data products accompanies this article.