Deep Neural Networks (DNNs) have emerged as the method of choice for solving a wide range of machine learning tasks. The enormous computational demand posed by DNNs is a key challenge for computing system designers and has most commonly been addressed through the design of DNN accelerators. However, these specialized accelerators utilize large quantities of multiply-accumulate units and on-chip memory and are prohibitive in area and cost constrained systems such as wearable devices and IoT sensors. In this work, we take a complementary approach and improve the performance of DNNs on general-purpose processor (GPP) cores. We do so by exploiting a key attribute of DNNs, viz. sparsity or the prevalence of zero values. We propose Sparsity-aware Core Extensions (SparCE)-a set of low-overhead micro-architectural and ISA extensions that dynamically detect whether an operand (e.g., the result of a load instruction) is zero and subsequently skip a set of future instructions that use it. To maximize performance benefits, SparCE ensures that the instructions to be skipped are prevented from even being fetched, as squashing instructions comes with a penalty (e.g., a pipeline stall). SparCE consists of 2 key micro-architectural enhancements. First, a Sparsity Register File (SpRF) is utilized to track registers that are zero. Next, a Sparsity-Aware Skip Address (SASA) Table is used to indicate instruction sequences that can be skipped, and to specify conditions on SpRF registers that trigger instruction skipping. When an instruction is fetched, SparCE dynamically pre-identifies whether the following instruction(s) can be skipped, and if so appropriately modifies the program counter, thereby skipping the redundant instructions and improving performance. We model SparCE using the gem5 architectural simulator, and evaluate our approach on 6 state-of-the-art image-recognition DNNs in the context of both training and inference using the Caffe deep learning framework. On a scalar microprocessor, SparCE achieves 1.11×-1.96× speedups across both convolution and fully-connected layers that exhibit 10-90 percent sparsity. These speedups translate to 19-31 percent reduction in execution time at the overall application-level. We also evaluate SparCE on a 4-way SIMD ARMv8 processor using the OpenBLAS library, and demonstrate that SparCE achieves 8-15 percent reduction in the application-level execution time.