The noise-induced diffusion of a particle in a two-dimensional (2D) periodic potential is studied by simulating a general-damping Langevin equation, where the particle is subjected especially to both a constant force along a direction and an Ornstein-Uhlenbeck (OU) colored noise. The effective diffusion coefficient is determined by the envelope width of the spatial distribution of the particle. The enhancement phenomenon of diffusion from fluctuation-controlled one to bias-plus-correlation-controlled one is investigated. A scheme of local harmonic approximation for the OU colored noise is used to analyze our numerical findings, showing that the correlation effect of noise is beneficial to biased diffusion. Moreover, the dimensional effect is discussed by comparing the 1D and 2D treatments. The prominent result is demonstrated in the tilted periodic potential, namely, the motion of particle exhibits two modes: the locked state and the running state, thus the effective diffusion coefficient can be enhanced if the probability of a particle locked in local minima of the potential increases or the space distribution of the particle can be disintegrated into many small pieces.