We study synchronization in delay-coupled oscillator networks using a master stability function approach. Within a generic model of Stuart-Landau oscillators (normal form of supercritical or subcritical Hopf bifurcation), we derive analytical stability conditions and demonstrate that by tuning the coupling phase one can easily control the stability of synchronous periodic states. We propose the coupling phase as a crucial control parameter to switch between in-phase synchronization or desynchronization for general network topologies or between in-phase, cluster, or splay states in unidirectional rings. Our results are robust even for slightly nonidentical elements of the network.