Globular clusters (GCs) are bright objects that span a wide range of galactocentric distances, and are thus probes of the structure of dark matter (DM) haloes. In this work, we explore whether the projected radial profiles of GCs can be used to infer the structural properties of their host DM haloes. We use the simulated GC populations in a sample of 166 central galaxies from the $(34.4~\rm cMpc)^3$ periodic volume of the E-MOSAICS project. We find that more massive galaxies host stellar and GC populations with shallower density profiles that are more radially extended. In addition, the metal-poor GC subpopulations tend to have shallower and more extended profiles than the metal-rich subsamples, which we relate to the preferentially accreted origin of the metal-poor GCs. We find strong correlations between the slopes and effective radii of the radial profiles of the GC populations and the structural properties of the DM haloes, such as their power-law slopes, scale radii, and concentration parameters. Accounting for a dependence on the galaxy stellar mass decreases the scatter of the two-dimensional relations. This suggests that the projected number counts of GCs, combined with their galaxy mass, trace the density profile of the DM halo of their host galaxy. When applied to extragalactic GC systems, we recover the scale radii and the extent of the DM haloes of a sample of ETGs with uncertainties smaller than $0.2~\rm dex$. Thus, extragalactic GC systems provide a novel avenue to explore the structure of DM haloes beyond the Local Group.