Phase unwrapping is a critical step to obtaining a continuous phase distribution in optical phase measurements and coherent imaging techniques. Traditional phase-unwrapping methods are generally low performance due to significant noise or undersampling. This paper proposes a deep convolutional neural network (DCNN) with a weighted jump-edge attention mechanism, namely, VDE-Net, to realize effective and robust phase unwrapping. Experimental results revealed that the weighted jump-edge attention mechanism, which is first proposed and simple to calculate, is useful for phase unwrapping. The proposed algorithm outperformed other networks or common attention mechanisms. In addition, an unseen wrapped phase image of a living red blood cell (RBC) was successfully unwrapped by the trained VDE-Net, thereby demonstrating its strong generalization capability.