Prediction of drug combinations that effectively target cancer cells is a critical challenge for cancer therapy, in particular for triple-negative breast cancer (TNBC), a highly aggressive breast cancer subtype with no effective targeted treatment. As signalling pathway networks critically control cancer cell behaviour, analysis of signalling network activity and crosstalk can help predict potent drug combinations and rational stratification of patients, thus bringing therapeutic and prognostic values. We have previously showed that the non-receptor tyrosine kinase PYK2 is a downstream effector of EGFR and c-Met and demonstrated their crosstalk signalling in basal-like TNBC. Here we applied a systems modelling approach and developed a mechanistic model of the integrated EGFR-PYK2-c-Met signalling network to identify and prioritize potent drug combinations for TNBC. Model predictions validated by experimental data revealed that among six potential combinations of drug pairs targeting the central nodes of the network, including EGFR, c-Met, PYK2 and STAT3, co-targeting of EGFR and PYK2 and to a lesser extent of EGFR and c-Met yielded strongest synergistic effect. Importantly, the synergy in co-targeting EGFR and PYK2 was linked to switch-like cell proliferation-associated responses. Moreover, simulations of patient-specific models using public gene expression data of TNBC patients led to predictive stratification of patients into subgroups displaying distinct susceptibility to specific drug combinations. These results suggest that mechanistic systems modelling is a powerful approach for the rational design, prediction and prioritization of potent combination therapies for individual patients, thus providing a concrete step towards personalized treatment for TNBC and other tumour types.