The Rényi entanglement entropy of quantum many-body systems can be viewed as the difference in free energy of partition functions with different trace topologies. We introduce an external field λ that controls the partition function topology, allowing us to define a notion of nonequilibrium work as λ is varied smoothly. Nonequilibrium fluctuation theorems of the work provide us with statistically exact estimates of the Rényi entanglement entropy. We put these ideas to use in the context of quantum Monte Carlo simulations of SU(N) symmetric spin models in one and two dimensions. In both cases we detect logarithmic violations to the area law with high precision, allowing us to extract the central charge for the critical 1D models and the number of Goldstone modes for the magnetically ordered 2D models.