Background: In recent years, stochastic descriptions of biochemical reactions based on the Master Equation (ME) have become widespread. These are especially relevant for models involving gene regulation. Gillespie’s Stochastic Simulation Algorithm (SSA) is the most widely used method for the numerical evaluation of these models. The SSA produces exact samples from the distribution of the ME for finite times. However, if the stationary distribution is of interest, the SSA provides no information about convergence or how long the algorithm needs to be run to sample from the stationary distribution with given accuracy. Results: We present a proof and numerical characterization of a Perfect Sampling algorithm for the ME of networks of biochemical reactions prevalent in gene regulation and enzymatic catalysis. Our algorithm combines the SSA with Dominated Coupling From The Past (DCFTP) techniques to provide guaranteed sampling from the stationary distribution. The resulting DCFTP-SSA is applicable to networks of reactions with uni-molecular stoichiometries and sub-linear, (anti-) monotone propensity functions. We showcase its applicability studying steady-state properties of stochastic regulatory networks of relevance in synthetic and systems biology. Conclusion: The DCFTP-SSA provides an extension to Gillespie’s SSA with guaranteed sampling from the stationary solution of the ME for a broad class of stochastic biochemical networks.