An efficient and accurate integration of stochastic (partial) differential equations with multiplicative noise can be obtained by separating the deterministic from the stochastic terms, the latter being treated by sampling exactly the solution of the associated Fokker-Planck equation. We demonstrate the computational power of this method by applying it to most absorbing phase transitions for which Langevin equations have been proposed. This provides precise estimates of the associated scaling exponents, clarifying the classification of these nonequilibrium problems, and confirms or refutes some existing theories.