  We study out-of-time order correlators (OTOCs) of the form $\langle\hat
A(t)\hat B(0)\hat C(t)\hat D(0)\rangle$ for a quantum system weakly coupled to
a dissipative environment. Such an open system may serve as a model of, e.g., a
small region in a disordered interacting medium coupled to the rest of this
medium considered as an environment. We demonstrate that for a system with
discrete energy levels the OTOC saturates exponentially $\propto \sum a_i
e^{-t/\tau_i}+const$ to a constant value at $t\rightarrow\infty$, in contrast
with quantum-chaotic systems which exhibit exponential growth of OTOCs.
Focussing on the case of a two-level system, we calculate microscopically the
decay times $\tau_i$ and the value of the saturation constant. Because some
OTOCs are immune to dephasing processes and some are not, such correlators may
decay on two sets of parametrically different time scales related to inelastic
transitions between the system levels and to pure dephasing processes,
respectively. In the case of a classical environment, the evolution of the OTOC
can be mapped onto the evolution of the density matrix of two systems coupled
to the same dissipative environment.
