Treatment of chronic myeloid leukemia (CML) with the tyrosine kinase inhibitor imatinib represents a successful application of molecularly targeted cancer therapy. A rapid hematologic and cytogenetic response can be induced in the majority of people, even in advanced disease. However, complete eradication of malignant cells, which are characterized by the expression of the BCR-ABL1 fusion protein, is rare. Reasons for the persistence of the malignant clone are currently not known and provide a substantial challenge for clinicians and biologists. Based on a mathematical modeling approach that quantitatively explains a broad range of phenomena, we show for two independent datasets that clinically observed BCR-ABL1 transcript dynamics during imatinib treatment of CML can consistently be explained by a selective functional effect of imatinib on proliferative leukemia stem cells. Our results suggest the general potential of imatinib to induce a complete elimination of the malignant clone. Moreover, we predict that the therapeutic benefit of imatinib can, under certain circumstances, be accelerated by combination with proliferation-stimulating treatment strategies.