At rest, the brain is traversed by spontaneous functional connectivity patterns. Two hypotheses have been proposed for their origins: they may reflect a continuous stream of ongoing cognitive processes as well as random fluctuations shaped by a fixed anatomical connectivity matrix. Here we show that both sources contribute to the shaping of resting-state networks, yet with distinct contributions during consciousness and anesthesia. We measured dynamical functional connectivity with functional MRI during the resting state in awake and anesthetized monkeys. Under anesthesia, the more frequent functional connectivity patterns inherit the structure of anatomical connectivity, exhibit fewer small-world properties, and lack negative correlations. Conversely, wakefulness is characterized by the sequential exploration of a richer repertoire of functional configurations, often dissimilar to anatomical structure, and comprising positive and negative correlations among brain regions. These results reconcile theories of consciousness with observations of long-range correlation in the anesthetized brain and show that a rich functional dynamics might constitute a signature of consciousness, with potential clinical implications for the detection of awareness in anesthesia and brain-lesioned patients.