Aims. We present the first three-dimensional internal motions for individual stars in the Draco dwarf spheroidal galaxy. Methods. By combining first-epoch Hubble Space Telescope observations and second-epoch Gaia Data Release 2 positions, we measured the proper motions of 149 sources in the direction of Draco. We determined the line-of-sight velocities for a sub-sample of 81 red giant branch stars using medium resolution spectra acquired with the DEIMOS spectrograph at the Keck II telescope. Altogether, this resulted in a final sample of 45 Draco members with high-precision and accurate 3D motions, which we present as a table in this paper. Results. Based on this high-quality dataset, we determined the velocity dispersions at a projected distance of ∼120 pc from the centre of Draco to be σ_R = 11.0_−1.5^+2.1 km s⁻¹, σ_T = 9.9_−3.1^+2.3 km s⁻¹ and σ_LOS = 9.0_−1.1^+1.1 km s⁻¹ in the projected radial, tangential, and line-of-sight directions. This results in a velocity anisotropy β = 0.25_−1.38^+0.47 at r ≳ 120 pc. Tighter constraints may be obtained using the spherical Jeans equations and assuming constant anisotropy and Navarro-Frenk-White (NFW) mass profiles, also based on the assumption that the 3D velocity dispersion should be lower than ≈1/3 of the escape velocity of the system. In this case, we constrain the maximum circular velocity V_max of Draco to be in the range of 10.2−17.0 km s⁻¹. The corresponding mass range is in good agreement with previous estimates based on line-of-sight velocities only. Conclusions. Our Jeans modelling supports the case for a cuspy dark matter profile in this galaxy. Firmer conclusions may be drawn by applying more sophisticated models to this dataset and with new datasets from upcoming Gaia releases.