We study continuous-time consensus dynamics for multi-agent systems with undirected switching interaction graphs. We establish a necessary and sufficient condition for exponential asymptotic consensus based on the classical theory of complete observability. The proof is remarkably simple compared to similar results in the literature and the conditions for consensus are mild. This observability-based method can also be applied to the case where negatively weighted edges are present. Additionally, as a by-product of the observability based arguments, we show that the nodes' initial value can be recovered from the signals on the edges up to a shift of the network average.