In this work, we explore a new approach to controlling networks of inverters to achieve synchronization of phase angles, frequencies, and ensure convergence to a desired voltage magnitude. We design a synchronizing vector field for a network of interconnected inverters which induces consensus-like dynamics and admits a fully decentralized implementation. We show that the combination of the synchronizing feedback and a simple voltage control law ensures synchronization of the inverters' phase angles and frequencies, as well as convergence of the voltage magnitudes from almost all initial conditions. Furthermore, we show that the controller exhibits a droop-like behavior around the standard operating point thus making it backwards-compatible with the existing power grid.