Exploring the climate impacts of various anthropogenic emissions scenarios is key to making informed decisions for climate change mitigation and adaptation. State-of-the-art Earth system models can provide detailed insight into these impacts, but have a large associated computational cost on a per-scenario basis. This large computational burden has driven recent interest in developing cheap machine learning models for the task of climate model emulation. In this manuscript, we explore the efficacy of randomly wired neural networks for this task. We describe how they can be constructed and compare them to their standard feedforward counterparts using the ClimateBench dataset. Specifically, we replace the dense layers in multilayer perceptrons, convolutional neural networks, and convolutional long short-term memory networks with randomly wired ones and assess the impact on model performance for models with 1 million and 10 million parameters. We find average performance improvements of 4.2