High levels of exozodiacal dust have been observed in the inner regions of a large fraction of main sequence stars. Given the short lifetime of the observed small dust grains, these 'exozodis' are difficult to explain, especially for old (>100 Myr) stars. The exozodiacal dust may be observed as excess emission in the mid-infrared, or using interferometry. We hypothesise that exozodi are sustained by planetesimals scattered by planets inwards from an outer planetesimal belt, where collision timescales are long. In this work, we use N-body simulations to show that the outwards migration of a planet into a belt, driven by the scattering of planetesimals, can increase, or sustain, the rate at which planetesimals are scattered from the outer belt to the exozodi region. We hypothesise that this increase is sufficient to sustain the observed exozodi on Gyr timescales. No correlation between observations of an outer belt and an exozodi is required for this scenario to work, as the outer belt may be too faint to detect. If planetesimal driven migration does explain the observed exozodi, this work suggests that the presence of an exozodi indicates the presence of outer planets and a planetesimal belt.Read less <