In the past decade, Deep Neural Networks (DNNs) achieved state-of-the-art performance in a broad range of problems, spanning from object classification and action recognition to smart building and healthcare. The flexibility that makes DNNs such a pervasive technology comes at a price: the computational requirements preclude their deployment on most of the resource-constrained edge devices available today to solve real-time and real-world tasks. This paper introduces a novel approach to address this challenge by combining the concept of predefined sparsity with Split Computing (SC) and Early Exit (EE). In particular, SC aims at splitting a DNN with a part of it deployed on an edge device and the rest on a remote server. Instead, EE allows the system to stop using the remote server and rely solely on the edge device's computation if the answer is already good enough. Specifically, how to apply such a predefined sparsity to a SC and EE paradigm has never been studied. This paper studies this problem and shows how predefined sparsity significantly reduces the computational, storage, and energy burdens during the training and inference phases, regardless of the hardware platform. This makes it a valuable approach for enhancing the performance of SC and EE applications. Experimental results showcase reductions exceeding 4x in storage and computational complexity without compromising performance. The source code is available at https://github.com/intelligolabs/sparsity_sc_ee.