We theoretically investigate current-induced magnetization dynamics in chiral-lattice helimagnets. Spin-orbit coupling in non-centrosymmetric crystals induces a reactive spin-transfer torque that has not been previously considered. We demonstrate how the torque is governed by the crystal symmetry and acts as an effective magnetic field along the current direction in cubic B20-type crystals. The effects of the new torque are computed for current-induced dynamics of spin-spirals and the Doppler shift of spin waves. In current-induced spin-spiral motion, the new torque tilts the spiral structure. The spin waves of the spiral structure are initially displaced by the new torque, while the dispersion relation is unaffected. ; Comment: Final version accepted by Physical Review B