Earth System Models (ESMs) are the primary tool used for understanding the global climate system and predicting its future evolution under anthropogenic forcing. However, these models are computationally very expensive, a problem especially acute for the simulations that underpin IPCC assessments of climate change. Before such simulations can be performed, ESMs must be “spun-up” to a stable, quasi-periodic pre-industrial state so that the impact of human forcing can be accurately determined. Such “spin-up” runs require several thousand years of simulation. This is due to the slow adjustment time scale of the ocean and terrestrial carbon cycle. Even on some of the world’s most powerful supercomputers, a single spin-up can take over 2 years of compute time. Besides the enormous cost in time and resources, this has important scientific and policy implications as it is prohibitively expensive to perform more than one such spin-up, increase resolution or propagate the large parametric uncertainty inherent in all ESMs into future projections. A robust and efficient solution to this so-called “spin-up problem” has long proved elusive. Here, I present a new approach based on Anderson Acceleration (AA) that is up to 10 times faster than conventional direct integration. A particular advantage of AA over methods that have been previously proposed, such as matrix-free Newton Krylov, is that it is entirely black-box, preserves conservation properties of the model, and is fully consistent with the models’ numerical time-stepping scheme. I will also describe Matlab and python implementations with checkpointing and restart capabilities, that are tailored for the batch HPC systems on which ESMs are typically run.