Synchronization of pulse-coupled oscillators (PCOs) has gained significant attention recently due to increased applications in sensor networks and wireless communications. Given the distributed and unattended nature of wireless sensor networks, it is imperative to enhance the resilience of pulse-based synchronization against malicious attacks. However, most existing results on resilient PCO synchronization are obtained for all-to-all networks. We propose a new pulse-based synchronization mechanism to improve the resilience of PCO synchronization that is applicable under general connected topologies. Under the proposed synchronization mechanism, we rigorously characterize the condition for stealthy Byzantine attacks and prove that perfect synchronization of legitimate oscillators can be guaranteed in the presence of multiple stealthy Byzantine attackers, irrespective of whether the attackers collude with each other or not. The new mechanism can guarantee resilient synchronization even when the initial phases of legitimate oscillators are widely distributed in a half circle, which is in distinct difference from most existing attack-resilient synchronization algorithms (including the seminal paper from Lamport and Melliar-Smith 1985) that require a priori (almost) synchronization among legitimate oscillators. Numerical simulation results are given to confirm the theoretical results.