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We study the top-k set similarity search problem using semantic overlap. While vanilla overlap requires exact matches between set elements, semantic overlap allows elements that are syntactically different but semantically related to increase the overlap. The semantic overlap is the maximum matching score of a bipartite graph, where an edge weight between two set elements is defined by a user-defined similarity function, e.g., cosine similarity between embeddings. Common techniques like token indexes fail for semantic search since similar elements may be unrelated at the character level. Further, verifying candidates is expensive (cubic versus linear for syntactic overlap), calling for highly selective filters. We propose Koios, the first exact and efficient algorithm for semantic overlap search. Koios leverages sophisticated filters to minimize the number of required graph-matching calculations. Our experiments show that for medium to large sets less than 5% of the candidate sets need verification, and more than half of those sets are further pruned without requiring the expensive graph matching. We show the efficiency of our algorithm on four real datasets and demonstrate the improved result quality of semantic over vanilla set similarity search.