Entropic causal inference is a recent framework for learning the causal graph between two variables from observational data for models with small entropy. In this paper, we first extend the causal graph identifiability result in the two-variable setting under relaxed assumptions. Next, we show the first identifiability result using the entropic approach for learning causal graphs with more than two nodes. We provide a sequential peeling algorithm that is provably correct for trees and complete graphs. We also propose a heuristic algorithm for small graphs. We conduct rigorous experiments that demonstrate performance of our algorithms under synthetic data with different generative models, as well as with real-world data.