This work investigates continuous time stochastic differential games with a large number of players, whose costs and dynamics interact through the empirical distribution of both their states and their controls. The control processes are assumed to be open-loop. We give regularity conditions guaranteeing that if the finite-player game admits a Nash equilibrium, then both the sequence of equilibria and the corresponding states processes satisfy a Sanov-type large deviation principle. The result requires existence of a Lipschitz continuous solution of the master equation of the corresponding mean field game, and is based on concentration inequalities for Lipschitz FBSDEs. The result carries over to cooperative (i.e. central planner) games. We study the linear-quadratic case of such games in details.