We study electroweakly interacting particles in rotating matter. The existence of the electric current along the axis of the matter rotation is predicted in this system. This new galvano-rotational effect is caused by the parity violating interaction between massless charged particles in the rotating matter. We start with the exact solution of the Dirac equation for a fermion involved in the electroweak interaction in the rotating frame. This equation includes the noninertial effects. Then, using the obtained solution, we derive the induced electric current which turns out to flow along the rotation axis. We study the possibility of the appearance of the galvano-rotational effect in dense matter of compact astrophysical