With the rapid advancement of automobile industry, autonomous driving in connected vehicles are expected to be the key technology to satisfy the expansion of human demands on more comfortable and safer driving experience. However, only on-board computation resources are insufficient to satisfy tough computation requirements of achieving full or even high automation. Therefore, autonomous driving with cloud/edge participation is desirable. In this paper, we propose E-Auto, a novel communication scheme to enable fast, stable, and accurate edge-assisted autonomous driving service for connected vehicles within any road types (e.g., driving on highway with very high speed or local roads with slow speed due to traffic congestion). In addition, as two key components of the proposed E-Auto scheme, a service period allocation algorithm and a frame resolution selection algorithm are designed to guarantee a sufficient frame rate for connected vehicles acquiring either uplink application (offload camera captured frames to the edge server) or downlink application (download entertainment videos). Through network simulations, we evaluate the performance of the proposed E-Auto scheme. Simulation results demonstrate that E-Auto can provide a high frame rate and low energy consumption autonomous driving service for connected vehicles.