Mutations in SWI/SNF genes are amongst the most common across all human cancers, but efficient therapeutic approaches that exploit vulnerabilities caused by SWI/SNF mutations are currently lacking. Here, we show that the SWI/SNF ATPases BRM/SMARCA2 and BRG1/SMARCA4 promote the expression of p62/GTF2H1, a core subunit of the transcription factor IIH (TFIIH) complex. Inactivation of either ATPase subunit downregulates GTF2H1 and therefore compromises TFIIH stability and function in transcription and nucleotide excision repair (NER). We also demonstrate that cells with permanent BRM or BRG1 depletion have the ability to restore GTF2H1 expression. As a consequence, the sensitivity of SWI/SNF-deficient cells to DNA damage induced by UV irradiation and cisplatin treatment depends on GTF2H1 levels. Together, our results expose GTF2H1 as a potential novel predictive marker of platinum drug sensitivity in SWI/SNF-deficient cancer cells.