  We show that dark matter substructure in galaxy-scale halos perturbs the time
delays between images in strong gravitational lens systems. The variance of the
effect depends on the subhalo mass function, scaling as the product of the
substructure mass fraction and a characteristic mass of subhalos (namely
<m^2>/<m>). Time delay perturbations therefore complement gravitational lens
flux ratio anomalies and astrometric perturbations by measuring a different
moment of the subhalo mass function. Unlike flux ratio anomalies, "time delay
millilensing" is unaffected by dust extinction or stellar microlensing in the
lens galaxy. Furthermore, we show that time delay ratios are immune to the
radial profile degeneracy that usually plagues lens modeling. We lay out a
mathematical theory of time delay perturbations and find it to be tractable and
attractive. We predict that in "cusp" lenses with close triplets of images,
substructure may change the arrival-time order of the images (compared with
smooth models). We discuss the possibility that this effect has already been
observed in RX J1131-1231.
