Angiogenesis is one of the hallmarks of cancer and, as such, one of the alternative general targets for anticancer therapy. Since angiogenesis is a complex process involving a high number of interconnected components, a network approach would be a convenient systemic way to analyse responses to directed drug attacks. Herein we show that, although the angiogenic network is easily broken by short combinations of directed attacks, it still remains essentially functional by keeping the global patterns and local efficiency essentially unaltered after these attacks. This is a clear sign of its high robustness and resilience and stresses the need of directed, combined attacks for an effective blockade of the process. The results of this theoretical study could be relevant for the design of new antiangiogenic therapies and the selection of their targets.