Balancing mutually diverging performance metrics, such as end-to-end latency, accuracy, and device energy consumption, is a challenging undertaking for deep neural network (DNN) inference in Just-in-Time edge environments that are inherently resource-constrained and loosely coupled. In this paper, we design and develop the Infer-EDGE framework that seeks to strike such a balance for latency-sensitive video processing applications. First, using comprehensive benchmarking experiments, we develop intuitions about the trade-off characteristics, which are then used by the framework to develop an Advantage Actor-Critic (A2C) Reinforcement Learning (RL) approach that can choose optimal run-time DNN inference parameters aligning the performance metrics based on the application requirements. Using real-world DNNs and a hardware testbed, we evaluate the benefits of the Infer-EDGE framework in terms of device energy savings, inference accuracy improvement, and end-to-end inference latency reduction.