Recent advances in deep neural networks (DNNs) have substantially improved the accuracy of intelligent applications. One effective scheme known as DNN partition further improves the speed of the inference by partitioning the DNN to a mobile device and its connected edge server to jointly process the inference. However, one of the challenges is how to maintain the service during handovers to avoid interruptions. Inspired by the recently developed early exit technique, where the DNN inference can be accelerated by leaving at an earlier exit point, we propose eDeepSave, a promising solution to save a large portion of video frames that cannot be handled during handovers. eDeepSave comprises three subschemes: (1) save the partially completed frames that are affected when the handover begins. (2) determine which frames we should save during a handover to maximize the number of saved frames. (3) repartition the last arriving frame before the end of the handover with a provable performance bound so that the frames after the handover can be processed without experiencing congestion. We build up a real-world prototype for the field experiments and extensive simulations, showing that eDeepSave can save up to 100% of the affected frames during handover.