We analyze a natural extension of the backward Euler approximation for a class of BSDEs with Lipschitz generators and random (unbounded) time horizons. We derive strong error bounds in terms of the underlying stepsize; the distance between the continuous terminal time and a discrete-time approximation; the distance between the terminal condition and a respective approximation; and an integrated distance depending on an approximation of the time component of the generator - all are scaled by the exponential of the maximal terminal time. As application we consider decoupled FBSDEs on bounded domains. We use an Euler-Maruyama scheme to approximate the diffusion and further refine our error bounds to only depend on the distance of the exit times.