The Carina Nebula is an active star-forming region with several open clusters rich in massive OB stars, thus making it an optimal target for studying stellar properties such as rotation for large samples of these early-type stars. We studied a sample of early-type stars probable members of the eight open clusters in the Carina Complex. The observational data consist of high-resolution spectra from the Gaia-ESO public Spectroscopic Survey. Astrometric and photometric data from Gaia EDR3 and radial velocities measured from the observed spectra are used to confirm the cluster members. The projected rotational velocities of 330 early-type stars of Carina are derived from the widths of He I lines at 4388 and 4471 Å. The reported $V \sin {i}$ values are the first estimates for 222 early-type stars. The $V \sin {i}$ distribution for the Carina clusters peaks at $\sim$100–150 km s$^{-1}$, consistent with the distributions for B stars in Galactic clusters. $V \sin {i}$ estimates for stars members of the clusters Trumpler 15, Collinder 228, Collinder 232, and Bochum 11 are presented for the first time in the literature. For a subsample of stars with earlier spectral types from B0 to B3, we find a bimodal distribution, with a third, small peak towards the upper values of $V \sin {i}$. When the full sample is split according to the parent cluster, we find that the oldest cluster in our sample, NGC 3293, presents a higher concentration of rapidly rotating stars. In contrast, Collinder 228 presents a larger number of stars with lower $V \sin {i}$.