Polo-like kinase 4 is essential for centrosome duplication, but its hyperactivation causes supernumerary centrosomes. Here we report that polo-like kinase 4 is directly phosphorylated and activated by stress-activated protein kinase kinase kinases (SAPKKKs). Stress-induced polo-like kinase 4 activation promotes centrosome duplication, whereas stress-induced SAPK activation prevents centrosome duplication. In the early phase of stress response, the balance of these opposing signals prevents centrosome overduplication. However, in the late phase of stress response, p53 downregulates polo-like kinase 4 expression, thereby preventing sustained polo-like kinase 4 activity and centrosome amplification. If both p53 and the SAPKK MKK4 are simultaneously inactivated, as is frequently found in cancer cells, persistent polo-like kinase 4 activity combined with the lack of SAPK-mediated inhibition of centrosome duplication conspire to induce supernumerary centrosomes under stress. Indeed, tumour-derived MKK4 mutants induced centrosome amplification under genotoxic stress, but only in p53-negative cells. Thus, our results reveal a mechanism that preserves the numeral integrity of centrosomes, and an unexplored tumour-suppressive function of MKK4.