In this paper we estimate the sparse dependence structure in the tail region of a multivariate random vector, potentially of high dimension. The tail dependence is modeled via a graphical model for extremes embedded in the H¨usler-Reiss distribution (Engelke and Hitz, 2020). We propose the extreme graphical lasso procedure to estimate the sparsity in the tail dependence, similar to the Gaussian graphical lasso method in high dimensional statistics. We prove its consistency in identifying the graph structure and estimating model parameters. The efficiency and accuracy of the proposed method are illustrated in simulated and real examples. (joint with Phyllis Wan)