Machine learning has become essential for automated classification of astronomical transients, but current approaches face significant limitations: classifiers trained on simulations struggle with real data, models developed for one survey cannot be easily applied to another, and new surveys require prohibitively large amounts of labelled training data. These challenges are particularly pressing as we approach the era of the Vera C. Rubin Observatory's Legacy Survey of Space and Time (LSST), where existing classification models will need to be retrained using LSST observations. We demonstrate that transfer learning can overcome these challenges by repurposing existing models trained on either simulations or data from other surveys. Starting with a model trained on simulated Zwicky Transient Facility (ZTF) light curves, we show that transfer learning reduces the amount of labelled real ZTF transients needed by 95% while maintaining equivalent performance to models trained from scratch. Similarly, when adapting ZTF models for LSST simulations, transfer learning achieves 94% of the baseline performance while requiring only 30% of the training data. These findings have significant implications for the early operations of LSST, suggesting that reliable automated classification will be possible soon after the survey begins, rather than waiting months or years to accumulate sufficient training data.