Type Ia supernovae (SNe Ia) play a critical role in astrophysics, yet their origin remains mysterious. A crucial physical mechanism in any SN Ia model is the initiation of the detonation front which ultimately unbinds the white dwarf progenitor and leads to the SN Ia. We demonstrate, for the first time, how a carbon detonation may arise in a realistic three-dimensional turbulent electron-degenerate flow, in a new mechanism we refer to as turbulently-driven detonation. Using both analytic estimates and three-dimensional numerical simulations, we show that strong turbulence in the distributed burning regime gives rise to intermittent turbulent dissipation which locally enhances the nuclear burning rate by orders of magnitude above the mean. This turbulent enhancement to the nuclear burning rate leads in turn to supersonic burning and a detonation front. As a result, turbulence plays a key role in preconditioning the carbon-oxygen fuel for a detonation. The turbulently-driven detonation initiation mechanism leads to a wider range of conditions for the onset of carbon detonation than previously thought possible, with important ramifications for SNe Ia models.