In this paper, we consider the synchronization control problem for uncertain high-order nonlinear multi-agent systems in a leader-follower scheme, under a directed communication protocol. A robust decentralized control law of minimal complexity is proposed that achieves prescribed, arbitrarily fast and accurate synchronization of the following agents with the leader. The control protocol is decentralized in the sense that the control signal of each agent is calculated based solely on local relative state information from its neighborhood set. Additionally, no information regarding the agents' dynamic model is employed in the design procedure. Moreover, provided that the communication graph is connected and contrary to the related works on multi-agent systems, the controller-imposed transient and steady state performance bounds are fully decoupled from: 1) the underlying graph topology, 2) the control gains selection, and 3) the agents' model uncertainties, and are solely prescribed by certain designer-specified performance functions. Extensive simulation results clarify and verify the approach.