Accurate distances are key to obtaining intrinsic properties of astronomical objects such as luminosity or size. Globular clusters (GCs) follow a well-defined relation between their absolute magnitudes and internal stellar velocity dispersions (σ), offering an independent way to measure distances to their host galaxies via high-resolution spectroscopy. This is reminiscent of the 'Faber-Jackson' for elliptical galaxies. However, unlike galaxies, GCs have a very narrow range of mass-to-light ratios and simple star formation histories. Here, we show that the GC MV-log10(σ) relation is linear, whose slope is identical for the Milky Way and M31 GC systems. Based on this, we use 94 Milky Way GCs which have distances from GAIA parallaxes, or proper-motion dispersion profiles to derive a 'GC velocity dispersion' distance (GCVD) to M31, obtaining (m - M)0 = 24.51 ± 0.08 (d = 798 ± 28 kpc), in excellent agreement with independent measurements. Combining data for these two galaxies to create a fiducial relation using 296 GCs with high-quality measurements, we obtain a zero-point uncertainty (±0.06 mag) corresponding to a distance uncertainty of $\sim 3~{{\ \rm per\ cent}}$. We then use GCVD to obtain a distance to the giant elliptical galaxy NGC 5128 (Centaurus A), finding (m - M)0 = 27.95 ± 0.09 (d = 3.89 ± 0.16 Mpc). This is in excellent agreement with, and in some cases more precise than, literature estimates from the tip of the red giant branch or surface brightness fluctuations. We apply GCVD to Local Group galaxies with appropriate data and find good agreement with literature values even in cases with only one GC velocity dispersion measurement.