Gene amplification drives oncogenesis in a broad spectrum of cancers. A number of drugs have been developed to inhibit the protein products of amplified driver genes, but their clinical efficacy is often hampered by drug resistance. Here, we introduce a therapeutic strategy for targeting cancer-associated gene amplifications by activating the DNA damage response with triplex-forming oligonucleotides (TFOs), which drives induction of apoptosis in tumors, whereas cells without amplifications process lower levels of DNA damage. Focusing on cancers driven by HER2-amplification, we find that TFOs targeting HER2 induce copy number-dependent DNA double strand breaks and activate p53-independent apoptosis in HER2-positive cancer cells and human tumor xenografts via a mechanism that is independent of HER2 cellular function. This strategy has demonstrated in vivo efficacy comparable with current precision medicines and provided a feasible alternative to combat drug resistance in HER-positive breast and ovarian cancer models. These findings offer a general strategy for targeting tumors with amplified genomic loci.