We mainly study the existence of random uniform exponential attractors for non-autonomous stochastic Schrödinger lattice system with multiplicative white noise and quasi-periodic forces in weighted spaces. Firstly, the stochastic Schrödinger system is transformed into a random system without white noise by the Ornstein-Uhlenbeck process, whose solution generates a jointly continuous non-autonomous random dynamical system $\mathrm{\Phi }$. Secondly, we prove the existence of a uniform absorbing random set for $\mathrm{\Phi }$ in weighted spaces. Finally, we obtain the existence of a random uniform exponential attractor for the considered system $\mathrm{\Phi }$ in weighted space.