In this paper, we consider a system of forward-backward stochastic differential equations (FBSDEs) with monotone functionals. We show the existence and uniqueness of such a system by the method of continuation similarly to Peng and Wu (1999) for classical FBSDEs and obtain estimates under conditional probability. As applications, we prove the well-posedness result for a mean field FBSDE with conditional law and show the existence of a decoupling function. In addition, we show that mean field games with common noise are uniquely solvable under a linear controlled process with convex and weak-monotone cost functions and prove that the optimal control is in a feedback form depending only on the current state and conditional law.