In large-scale cloud service systems, support tickets serve as a critical mechanism for resolving customer issues and maintaining service quality. However, traditional manual ticket escalation processes encounter significant challenges, including inefficiency, inaccuracy, and difficulty in handling the high volume and complexity of tickets. While previous research has proposed various machine learning models for ticket classification, these approaches often overlook the practical demands of real-world escalations, such as dynamic ticket updates, topic-specific routing, and the analysis of ticket relationships. To bridge this gap, this paper introduces TickIt, an innovative online ticket escalation framework powered by Large Language Models. TickIt enables topic-aware, dynamic, and relationship-driven ticket escalations by continuously updating ticket states, assigning tickets to the most appropriate support teams, exploring ticket correlations, and leveraging category-guided supervised fine-tuning to continuously improve its performance. By deploying TickIt in ByteDance's cloud service platform Volcano Engine, we validate its efficacy and practicality, marking a significant advancement in the field of automated ticket escalation for large-scale cloud service systems.