Entanglement is the fundamental difference between classical and quantum systems and has become one of the guiding principles in the exploration of high- and low-energy physics. The calculation of entanglement entropies in interacting quantum field theories, however, remains challenging. Here, we present the first method for the explicit computation of reduced density matrices of interacting quantum field theories using truncated Hamiltonian methods. The method is based on constructing an isomorphism between the Hilbert space of the full system and the tensor product of Hilbert spaces of subintervals. This naturally enables the computation of the von Neumann and arbitrary Rényi entanglement entropies as well as mutual information, logarithmic negativity, and other measures of entanglement. Our method is applicable to equilibrium states and nonequilibrium evolution in real time. It is model independent and can be applied to any Hamiltonian truncation method that uses a free basis expansion. We benchmark the method on the free Klein-Gordon theory finding excellent agreement with the analytic results. We further demonstrate its potential on the interacting sine-Gordon model, studying the scaling of von Neumann entropy in ground states and real-time dynamics following quenches of the model.