Hypervelocity stars (HVSs) travel from the Galactic Centre across the dark matter halo of the Milky Way, where they are observed with velocities in excess of the Galactic escape speed. Because of their quasi-radial trajectories, they represent a unique probe of the still poorly constrained dark matter component of the Galactic potential. In this paper, we present a new method to produce such constraints. Our likelihood is based on the local HVS density obtained by back-propagating the observed phase space position and quantifies the ejection probability along the orbit. To showcase our method, we apply it to simulated Gaia samples of $sim200$ stars in three realistic Galactic potentials with dark matter components parametrized by spheroidal NFW profiles. We find that individual HVSs exhibit a degeneracy in the scale mass-scale radius plane ($M_s-r_s$) and are able to measure only the combination $alpha = M_s/r_s^2$. Likewise, a degeneracy is also present between $alpha$ and the spheroidal axis-ratio $q$. In the absence of observational errors, we show the whole sample can nail down both parameters with {it sub-per cent} precision (about $1%$ and $0.1%$ for $alpha$ and $q$ respectively) with no systematic bias. This remarkable power to constrain deviations from a symmetric halo is a consequence of the Galactocentric origin of HVSs. To compare our results with other probes, we break the degeneracy in the scale parameters and impose a mass-concentration relation. The result is a competitive precision on the virial mass $M_{200}$ of about $10%$.
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