  The cosmological memory effect is a permanent change in the relative
separation of test particles located in a FLRW spacetime due to the passage of
gravitational waves. In the case of a spatially flat FLRW spacetime filled with
a perfect fluid in general relativity, it is known that only tensor
perturbations contribute to the memory effect while scalar and vector
perturbations do not. In this paper, we show that in the context of
scalar-tensor theories, the scalar perturbations associated to the scalar
graviton contribute to the memory effect as well. We find that, depending on
the mass and coupling, the influence of cosmic expansion on the memory effect
due to the scalar perturbations can be either stronger or weaker than the one
induced by the tensor perturbations. As a byproduct, in an appendix, we develop
a general framework which can be used to study coupled wave equations in any
curved spacetime region which admits a foliation by time slices.
