We study the Sivers asymmetry in semi-inclusive deep inelastic scattering: the three-gluon correlation function contribution. At the cross section level, we establish the matching between the twist-3 collinear factorization approach and the transverse momentum dependent factorization formalism at the moderate transverse momentum region. The so-called coefficient functions when one expands the quark Sivers function in terms of three-gluon correlation functions are derived, which are essential components in the usual transverse momentum dependent evolution formalism. Finally we calculate the transverse-momentum-weighted spin-dependent differential cross section at the next-to-leading order, from which we identify the off-diagonal/three-gluon correlation contribution to the QCD collinear evolution of the twist-3 Qiu-Sterman function.