We use Gaia DR2 systemic proper motions of 45 satellite galaxies to constrain the mass of the Milky Way using the scale-free mass estimator of Watkins et al. (2010). We first determine the anisotropy parameter β, and the tracer satellites' radial density index γ to be β = $-0.67^{+0.45}_{-0.62}$ and γ = 2.11 ± 0.23. When we exclude possible former satellites of the Large Magellanic Cloud, the anisotropy changes to β = $-0.21^{+0.37}_{-0.51}$ . We find that the index of the Milky Way's gravitational potential α, which is dependent on the mass itself, is the parameter with the largest impact on the mass determination. Via comparison with cosmological simulations of Milky Way-like galaxies, we carried out a detailed analysis of the estimation of the observational uncertainties and their impact on the mass estimator. We found that the mass estimator is biased when applied naively to the satellites of simulated Milky Way haloes. Correcting for this bias, we obtain for our Galaxy a mass of $0.58^{+0.15}_{-0.14}\times 10^{12}$ M☉ within 64 kpc, as computed from the inner half of our observational sample, and $1.43^{+0.35}_{-0.32}\times 10^{12}$ M☉ within 273 kpc, from the full sample; this latter value extrapolates to a virial mass of $M_\mathrm{vir\, \Delta =97}=1.51^{+0.45}_{-0.40} \times 10^{12}\,{\rm M}_{\odot }$ corresponding to a virial radius of Rvir = 308 ± 29 kpc. This value of the Milky Way mass lies in-between other mass estimates reported in the literature, from various different methods.