This paper investigates the synchronization problem of nonlinear multi-agent systems with time-varying control in the presence of transmission delay over a communication network. To facilitate the study, a novel delayed differential inequality with time-varying coefficients is first established. Then, the synchronization problem is recast into the stability problem of a delayed differential system with certain time-dependent parameter. A sufficient criterion is further formulated to guarantee the synchronization. The mathematical proof reveals that the time average of the control strength is crucial for reaching synchronization. Together with the agent dynamics and the topology, this average also governs the largest admissible delay. Moreover, the criterion is applied to synchronization problems with general on-off coupling under data sampling and delayed communications, respectively. Some useful corollaries are consequently deduced. Finally, numerical simulations are presented to illustrate the validity of our theoretical results.