Consider the underdamped Langevin process $(q(t),p(t))_{t\geq0}$ in $\R^d\times\R^d$. We derive the low-temperature asymptotic of its mean-transition time between basins of attraction for a double-well potential. This asymptotic is called Eyring-Kramers law and often relies in the literature on Potential theory tools which are ill-defined for hypoelliptic processes like the underdamped Langevin process. In this work, we implement a novel approach which circumvents the use of these traditional methods.