A par at the first extra hole sealed victory for the 33-year-old, who won his first PGA Tour title since last year's US Open.
Rose had been ahead of Stefani on five under at his 72nd hole, but could only drive into the trees on the left.
They finished tied at four-under-par, but the American found a water hazard with his second shot at the Congressional.
"I think Congressional wanted its reputation back," Rose told BBC Sport. "It tested all of us getting to four-under par to win this tournament."
Victory for Rose sealed his sixth PGA Tour title.