The away side bagged their second goal 20 minutes later when Darren Bent tapped in from Young's unselfish pass.
Montenegro got themselves back into the match on half-time when Elsad Zverotic's deflected shot found the net.
In a scrappy second half, England were reduced to 10 men when Rooney went straight through the back of Miodrag Dzudovic after losing the ball to the defender.
The dismissal means that the striker will miss the first game of the finals, and could miss a second at the discretion of UEFA.
The hosts used their numerical advantage to increase the pressure on their opponents and were rewarded for their efforts deep into injury time when Andrija Delibasic headed home.
Montenegro have also made it through to the playoffs after Switzerland lost 2-0 to Wales.