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 $sim120$ pc from the centre of Draco to be $sigma_{R} =11.0^{+2.1}_{-1.5}$ km/s, $sigma_{T}=9.9^{+2.3}_{-3.1}$ km/s and $sigma_{LOS}=9.0^{+1.1}_{-1.1}$ km/s in the projected radial, tangential, and line-of-sight directions. This results in a velocity anisotropy $beta=0.25^{+0.47}_{-1.38}$ at $r gtrsim120$ 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 $approx 1/3$ 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. 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.
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