The order from quantum disorder (OFQD) effect was first discovered in frustrated quantum spin systems. Recently, OFQD was also found in Rashba spin-orbit-coupled (SOC) systems even on a square lattice. Here we study the role of OFQD in the experimentally realized weakly interacting bosonic quantum anomalous Hall system of spinor bosons in an optical lattice. We perform microscopic nonperturbative OFQD calculations and construct an effective action based on symmetry considerations. These two complementary approaches establish an OFQD-induced quantum critical regime at finite Zeeman field. In particular, we show that the competition between the OFQD-generated effective potential and the Zeeman field drives a continuous quantum phase transition from the canted-antiferromagnetic superfluid phase to the $Z$-ferromagnetic superfluid phase with dynamical exponent $z=2$. We further determine the associated finite-temperature quantum critical scaling and discuss experimentally relevant signatures in existing cold-atom platforms. Our results establish OFQD as a mechanism for an interaction-driven quantum phase transition in the bosonic quantum anomalous Hall system.