The growing vehicle connectivity and autonomy in the ground transportation system are not only able to improve traffic safety but also fuel efficiency. This paper proposes a computational friendly receding-horizon optimization-based nonlinear model predictive control (NMPC) algorithm to achieve fuel-saving speed planning for connected vehicles. The NMPC method solves for the fuel-optimal speed profile of connected vehicles considering a short speed preview of the preceding vehicle. By utilizing such previewing information through vehicle connectivity, the fuel consumption of the connected vehicles is reduced by avoiding unnecessary braking and acceleration, particularly in transient operating conditions. In order to analyze the effectiveness of NMPC design, dynamic programming (DP) method is adopted as a benchmark algorithm where the full speed preview of the preceding vehicle is known. The performances of NMPC and DP designs in driving behavior and fuel economy are quantitatively explored and compared under several standard driving cycles. Results show a promising improvement of the performance by adopting the proposed design and reveal the potential fuel benefits brought by vehicle connectivity and autonomy.