Web applications are gradually shifting toward resource-constrained mobile devices. As a result, the Web runtime system must simultaneously address two challenges: responsiveness and energy-efficiency. Conventional Web runtime systems fall short due to their reactive nature: they react to a user event only after it is triggered. The reactive strategy leads to local optimizations that schedule event executions one at a time, missing global optimization opportunities. This paper proposes Proactive Event Scheduling (PES). The key idea of PES is to proactively anticipate future events and thereby globally coordinate scheduling decisions across events. Specifically, PES predicts events that are likely to happen in the near future using a combination of statistical inference and application code analysis. PES then speculatively executes future events ahead of time in a way that satisfies the QoS constraints of all the events while minimizing the global energy consumption. Fundamentally, PES unlocks more optimization opportunities by enlarging the scheduling window, which enables coordination across both outstanding events and predicted events. Hardware measurements show that PES reduces the QoS violation and energy consumption by 61.2% and 26.5%, respectively, over the Android default Interactive CPU governor. It also reduces the QoS violation and energy consumption by 63.1% and 17.9%, respectively, compared to EBS, a state-of-the-art reactive scheduler.