During image editing, existing deep generative models tend to re-synthesize the entire output from scratch, including the unedited regions. This leads to a significant waste of computation, especially for minor editing operations. In this work, we present Spatially Sparse Inference (SSI), a general-purpose technique that selectively performs computation for edited regions and accelerates various generative models, including both conditional GANs and diffusion models. Our key observation is that users tend to make gradual changes to the input image. This motivates us to cache and reuse the feature maps of the original image. Given an edited image, we sparsely apply the convolutional filters to the edited regions while reusing the cached features for the unedited regions. Based on our algorithm, we further propose Sparse Incremental Generative Engine (SIGE) to convert the computation reduction to latency reduction on off-the-shelf hardware. With 1.2%-area edited regions, our method reduces the computation of DDIM by $7.5\times$ and GauGAN by $18\times$ while preserving the visual fidelity. With SIGE, we accelerate the inference time of DDIM by $3.0\times$ on RTX 3090 and $6.6\times$ on Apple M1 Pro CPU, and GauGAN by $4.2\times$ on RTX 3090 and $14\times$ on Apple M1 Pro CPU.