We apply recent developments in large momentum effective theory (LaMET) to formulate a nonperturbative calculation of the single transverse-spin asymmetry from the so-called Sivers mechanism in terms of quasi-transverse-momentum-dependent quark distribution functions. When the spin asymmetry is defined as the ratio of the quark Sivers function over the spin-averaged distribution, it can be directly calculated in terms of the relevant quasidistributions with the soft functions and perturbative matching kernels canceling out. Apart from the general formula presented, we have verified the result in the small transverse distance limit at one-loop order, which reduces to a collinear expansion at the twist-three level.