Some mutations in cancer cells can be exploited for therapeutic intervention. However, for many cancer subtypes, including triple‐negative breast cancer (TNBC), no frequently recurring aberrations could be identified to make such an approach clinically feasible. Characterized by a highly heterogeneous mutational landscape with few common features, many TNBCs cluster together based on their ‘basal‐like’ transcriptional profiles. We therefore hypothesized that targeting TNBC cells on a systems level by exploiting the transcriptional cell state might be a viable strategy to find novel therapies for this highly aggressive disease. We performed a large‐scale chemical genetic screen and identified a group of compounds related to the drug PKC412 (midostaurin). PKC412 induced apoptosis in a subset of TNBC cells enriched for the basal‐like subtype and inhibited tumor growth in vivo. We employed a multi‐omics approach and computational modeling to address the mechanism of action and identified spleen tyrosine kinase (SYK) as a novel and unexpected target in TNBC. Quantitative phosphoproteomics revealed that SYK inhibition abrogates signaling to STAT3, explaining the selectivity for basal‐like breast cancer cells. This non‐oncogene addiction suggests that chemical SYK inhibition may be beneficial for a specific subset of TNBC patients and demonstrates that targeting cell states could be a viable strategy to discover novel treatment strategies.