A number of astrophysical scenarios possess and preserve an overall cylindrical symmetry when also undergoing a catastrophic and nonlinear evolution. Exploiting such a symmetry, these processes can be studied through numerical-relativity simulations at smaller computational costs and at considerably larger spatial resolutions. We present here a new flux-conservative formulation of the relativistic hydrodynamics equations in cylindrical coordinates. By rearranging those terms in the equations which are the sources of the largest numerical errors, the new formulation yields a global truncation error, which is one or more orders of magnitude smaller than those of alternative and commonly used formulations. We illustrate this through a series of numerical tests involving the evolution of oscillating spherical and rotating stars, as well as shock-tube tests.