We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on R ^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale ε> 0 , we establish homogenization error estimates of the order ε in case d≧ 3 , and of the order ε| log ε| ^{1 / 2} in case d= 2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence ε ^δ. We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/ ε) ^{- d / 2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C ^{1 , α} regularity theory is available.