In this paper we present high-resolution VLT/FLAMES observations of red giant stars in the massive intermediate-age Large Magellanic Cloud star cluster NGC 1846, which, on the basis of its extended main-sequence turnoff (EMSTO), possesses an internal age spread of approximate to 300 Myr. We describe in detail our target selection and data reduction procedures, and construct a sample of 21 stars possessing radial velocities indicating their membership of NGC 1846 at high confidence. We consider high-resolution spectra of the planetary nebulaMo-17, and conclude that this object is also a member of the cluster. Our measured radial velocities allow us to conduct a detailed investigation of the internal kinematics of NGC 1846, the first time this has been done for an EMSTO system. The key result of this work is that the cluster exhibits a significant degree of systemic rotation, of a magnitude comparable to the mean velocity dispersion. Using an extensive suite of Monte Carlo models we demonstrate that, despite our relatively small sample size and the substantial fraction of unresolved binary stars in the cluster, the rotation signal we detect is very likely to be genuine. Our observations are in qualitative agreement with the predictions of simulations modeling the formation of multiple populations of stars in globular clusters, where a dynamically cold, rapidly rotating second generation is a common feature. NGC 1846 is less than one relaxation time old, so any dynamical signatures encoded during its formation ought to remain present.