Recent studies have shown that dark matter with a superfluid phase in which phonons mediate a long-distance force gives rise to the phenomenologically well-established regularities of Modified Newtonian Dynamics (MOND). Superfluid dark matter, therefore, has emerged as a promising explanation for astrophysical observations by combining the benefits of both particle dark matter and MOND, or its relativistic completions, respectively... We here investigate whether superfluid dark matter can reproduce the observed Milky Way rotation curve for $ R < 25\,\rm{kpc}$ and are able to answer this question in the affirmative. Our analysis demonstrates that superfluid dark matter fits the data well with parameters in reasonable ranges. The most notable difference between superfluid dark matter and MOND is that superfluid dark matter requires about $ 20\% $ less total baryonic mass (with a suitable interpolation function). The total baryonic mass is then $5.96 \cdot 10^{10}\,M_\odot$, of which $1.03\cdot10^{10}\,M_\odot$ are from the bulge, $3.95\cdot10^{10}\,M_\odot$ are from the stellar disk, and $0.98\cdot10^{10}\,M_\odot$ are from the gas disk. Our analysis further allows us to estimate the radius of the Milky Way's superfluid core (concretely, the so-called NFW and thermal radii) and the total mass of dark matter in both the superfluid and the normal phase. By varying the boundary conditions of the superfluid to give virial masses $M_{200}^{\rm{DM}}$ in the range $0.5-3.0\cdot10^{12}\,M_\odot$, we find that the NFW radius $R_{\rm{NFW}}$ varies between $65\,\rm{kpc}$ and $73\,\rm{kpc}$, while the thermal radius $R_T$ varies between about $67\,\rm{kpc}$ and $105\,\rm{kpc}$. This is the first such treatment of a non-spherically-symmetric system in superfluid dark matter. (read more)