Sprinting is a class of mechanisms that provides a short but significant performance boost while temporarily exceeding the thermal design point. We propose DynaSprint, a software runtime that manages sprints by dynamically predicting utility and modeling thermal headroom. Moreover, we propose a new sprint mechanism for caches, increasing capacity briefly for enhanced performance. For a system that extends last-level cache capacity from 2MB to 4MB per core and can absorb 10J of heat, DynaSprint-guided cache sprints improve performance by 17% on average and by up to 40% over a non-sprinting system. These performance outcomes, within 95% of an oracular policy, are possible because DynaSprint accurately predicts phase behavior and sprint utility.