In SU(3) lattice QCD formalism, we propose a method to extract gauge fields from link variables analytically. With this method, we perform the first study on effective mass generation of off-diagonal gluons and infrared Abelian dominance in the maximally Abelian gauge in the SU(3) case. Using SU(3) lattice QCD, we investigate the propagator and the effective mass of the gluon fields in the maximally Abelian gauge with U(1)(3) x U(1)(8) Landau gauge fixing. The Monte Carlo simulation is performed on 164 at beta = 5.7, 5.8, and 6.0 at the quenched level. The off-diagonal gluons behave as massive vector bosons with the approximate effective mass M-off similar or equal to 1.0-1.1 GeV in the region of r = 0.3-0.8 fm, and the propagation is limited within a short range, while the propagation of diagonal gluons remains even in a large range. In this way, infrared Abelian dominance is shown in terms of short-range propagation of off-diagonal gluons. Furthermore, we investigate the functional form of the off-diagonal gluon propagator. The functional form is well described by the four-dimensional Euclidean Yukawa-type function e(-moffr)/r with m(off) similar or equal to 1.2-1.3 GeV for r = 0.1-0.8 fm. This also indicates that the spectral function of off-diagonal gluons has the negative-value region.