Most UAVs depend on realtime 3D point clouds to navigate through unknown environments. However, both point clouds processing and trajectory planning are computationally expensive and will deplete UAV's battery quickly. There are also inevitable uncertainties in point clouds, which further makes collision-free trajectory planning a very challenging problem. To address these issues, we propose an energy-efficient and cloud-assisted autonomous navigation system, called ECONOMY, which allows a UAV to transmit the point clouds through a cellular network to a computing cloud that plans the trajectories for the UAV in realtime. To maximize the UAV's energy-efficiency, we jointly optimize its velocity, trajectory, and transmission power. Since the formulated problem is non-convex, we decompose it into a UAV trajectory and velocity optimization problem (UTVO), and a UAV communication optimization problem (UCO), which are resolved by an intelligent solver built upon the Suggest-and-Improve framework, and gradient ascent, respectively. To address point clouds uncertainties, we devise probabilistic collision-free constraints which can be handled in a deterministic manner by exploiting the tight $(n,2,\mathcal {R}^n)$-upper bound for convex sets. Simulation results for a UAV exploring a simulated urban environment demonstrate the efficacy and efficiency of ECONOMY.