As is well known, the quantum evolution speed of quantum state can never be accelerated in the Markovian regime without any operators on the system. The Hamiltonian corrections induced by the action of coherent driving forces are often used to fight dissipative and decoherence mechanisms in experiments. For this reason, considering three noisy channels (the phase-flip channel, the amplitude damping channel and the depolarizing channel), we propose a scheme of speedup evolution of an open system by controlling an external unitary coherent driving operator on the system. It is shown that, in the presence of the coherent driving, no-speedup evolution can be transformed into quantum speedup evolution in the Markovian dynamics process. Additionally, under the fixed coherent driving strength in the above three noisy channels, the best way to achieve the most degree of quantum speedup for the system has been acquired by rotating the system with appropriate driving direction angles, respectively. Finally, we conclude that the reason for this acceleration is not the non-Markovian dynamical behavior of the system but due to the oscillation of geometric distance between the initial state and the target final state.