We prove stochastic homogenization for integral functionals defined on Sobolev spaces, where the stationary, ergodic integrand satisfies a degenerate growth condition of the form c|xi A(w, x)|p 5 f (w, x, xi) 5 |xi A(w, x)|p + A(w, x) for some p is an element of (1, +infinity) and with a sta-tionary and ergodic diagonal matrix A such that its norm and the norm of its inverse satisfy minimal integrability assumptions and A is a nonnegative, stationary function with finite first moment. We also consider the convergence when Dirichlet boundary conditions or an obstacle condition are imposed. Assuming the strict convexity and differentiability of f with respect to its last variable, we further prove that the homogenized integrand is also strictly convex and differentiable. These properties allow us to show homogenization of the associated Euler-Lagrange equations.