Positive dependence is present in many real world data sets and has appealing stochastic properties that can be exploited in statistical modeling and in estimation. In particular, the notion of multivariate total positivity of order 2 ($ \mathrm{MTP}_{2} $) is a convex constraint and acts as an implicit regularizer in the Gaussian case. We study positive dependence in multivariate extremes and introduce $ \mathrm{EMTP}_{2} $, an extremal version of $ \mathrm{MTP}_{2} $. This notion turns out to appear prominently in extremes, and in fact, it is satisfied by many classical models. For a Hüsler--Reiss distribution, the analogue of a Gaussian distribution in extremes, we show that it is $ \mathrm{EMTP}_{2} $ if and only if its precision matrix is a Laplacian of a connected graph. We propose an estimator for the parameters of the Hüsler--Reiss distribution under $ \mathrm{EMTP}_{2} $ as the solution of a convex optimization problem with Laplacian constraint. We prove that this estimator is consistent and typically yields a sparse model with possibly nondecomposable extremal graphical structure. Applying our methods to a data set of Danube River flows, we illustrate this regularization and the superior performance compared to existing methods.