Diffusion models have achieved state-of-the-art results in generative modelling but remain computationally intensive at inference time, often requiring thousands of discretization steps. To this end, we propose Sig-DEG (Signature-based Differential Equation Generator), a novel generator for distilling pre-trained diffusion models, which can universally approximate the backward diffusion process at a coarse temporal resolution. Inspired by high-order approximations of stochastic differential equations (SDEs), Sig-DEG leverages partial signatures to efficiently summarize Brownian motion over sub-intervals and adopts a recurrent structure to enable accurate global approximation of the SDE solution. Distillation is formulated as a supervised learning task, where Sig-DEG is trained to match the outputs of a fine-resolution diffusion model on a coarse time grid. During inference, Sig-DEG enables fast generation, as the partial signature terms can be simulated exactly without requiring fine-grained Brownian paths. Experiments demonstrate that Sig-DEG achieves competitive generation quality while reducing the number of inference steps by an order of magnitude. Our results highlight the effectiveness of signature-based approximations for efficient generative modeling.