Criticality has gained widespread interest in neuroscience as an attractive framework for understanding the character and functional implications of variability in brain activity. The metastability of critical systems maximizes their dynamic range, storage capacity, and computational power. Power-law scaling—a hallmark of criticality—has been observed on different levels, e.g., in the distribution of neuronal avalanches in vitro and in vivo, but also in the decay of temporal correlations in behavioral performance and ongoing oscillations in humans. An unresolved issue is whether power-law scaling on different organizational levels in the brain—and possibly in other hierarchically organized systems—can be related. Here, we show that critical-state dynamics of avalanches and oscillations jointly emerge in a neuronal network model when excitation and inhibition is balanced. The oscillatory activity of the model was qualitatively similar to what is typically observed in recordings of human resting-state MEG. We propose that homeostatic plasticity mechanisms tune this balance in healthy brain networks, and that it is essential for critical behavior on multiple levels of neuronal organization with ensuing functional benefits. Based on our network model, we introduce a concept of multi-level criticality in which power-law scaling can emerge on multiple time scales in oscillating networks.