Cross-lingual transfer between a high-resource language and its dialects or closely related language varieties should be facilitated by their similarity, but current approaches that operate in the embedding space do not take surface similarity into account. In this work, we present a simple yet effective strategy to improve cross-lingual transfer between closely related varieties by augmenting the data of the high-resource parent language with character-level noise to make the model more robust towards spelling variations. Our strategy shows consistent improvements over several languages and tasks: Zero-shot transfer of POS tagging and topic identification between language varieties from the Germanic, Uralic, and Romance language genera. Our work provides evidence for the usefulness of simple surface-level noise in improving transfer between language varieties.