We report the discovery of 13 new pulsars in the globular cluster NGC 1851 by the TRAPUM Large Survey Project using the MeerKAT radio telescope. The discoveries consist of six isolated millisecond pulsars (MSPs) and seven binary pulsars, of which six are MSPs and one is mildly recycled. For all the pulsars, we present the basic kinematic, astrometric, and orbital parameters, where applicable, as well as their polarimetric properties, when these are measurable. Two of the binary MSPs (PSR J0514−4002D and PSR J0514−4002E) are in wide and extremely eccentric (e > 0.7) orbits with a heavy white dwarf and a neutron star as their companion, respectively. With these discoveries, NGC 1851 is now tied with M28 as the cluster with the third largest number of known pulsars (14). Its pulsar population shows remarkable similarities with that of M28, Terzan 5, and other clusters with comparable structural parameters. The newly found pulsars are all located in the innermost regions of NGC 1851 and will likely enable, among other things, detailed studies of the cluster structure and dynamics.