Reactive Oxygen Species (ROS) are a natural by-product of cellular growth and proliferation, and are required for fundamental processes such as protein-folding and signal transduction. However, ROS accumulation, and the onset of oxidative stress, can negatively impact cellular and genomic integrity. Signalling networks have evolved to respond to oxidative stress by engaging diverse enzymatic and non-enzymatic antioxidant mechanisms to restore redox homeostasis. The architecture of oxidative stress response networks during periods of normal growth, and how increased ROS levels dynamically reconfigure these networks are largely unknown. In order to gain insight into the structure of signalling networks that promote redox homeostasis we first performed genome-scale RNAi screens to identify novel suppressors of superoxide accumulation. We then infer relationships between redox regulators by hierarchical clustering of phenotypic signatures describing how gene inhibition affects superoxide levels, cellular viability, and morphology across different genetic backgrounds. Genes that cluster together are likely to act in the same signalling pathway/complex and thus make "functional interactions". Moreover we also calculate differential phenotypic signatures describing the difference in cellular phenotypes following RNAi between untreated cells and cells submitted to oxidative stress. Using both phenotypic signatures and differential signatures we construct a network model of functional interactions that occur between components of the redox homeostasis network, and how such interactions become rewired in the presence of oxidative stress. This network model predicts a functional interaction between the transcription factor Jun and the IRE1 kinase, which we validate in an orthogonal assay. We thus demonstrate the ability of systems-biology approaches to identify novel signalling events.