Classical quasi-integrable systems are known to have Lyapunov times much shorter than their ergodicity time—the clearest example being the Solar System—but the situation for their quantum counterparts is less well understood. As a first example, we examine the quantum Lyapunov exponent, defined by the evolution of the four-point out-of-time-order correlator (OTOC), of integrable systems which are weakly perturbed by an external noise, a setting that has proven to be illuminating in the classical case. In analogy to the tangent space in classical systems, we derive a linear superoperator equation which dictates the OTOC dynamics. (1) We find that in the semiclassical limit the quantum Lyapunov exponent is given by the classical one: it scales as ε1/3, with ε being the variance of the random drive, leading to short Lyapunov times compared to the diffusion time (which is ∼ε−1). (2) We also find that in the highly quantal regime the Lyapunov instability is suppressed by quantum fluctuations, and (3) for sufficiently small perturbations the ε1/3 dependence is also suppressed—another purely quantum effect which we explain. These essential features of the problem are already present in a rotor that is kicked weakly but randomly. Concerning quantum limits on chaos, we find that quasi-integrable systems are relatively good scramblers in the sense that the ratio between the Lyapunov exponent and kT/ℏ may stay finite at a low temperature T.