You are correct, here is the official ruling:

As long as the fielder is not touching the ground in dead ball territory when he catches the ball, it is a legal catch if he holds onto the ball and meets the definition of a catch. If the catch is not the third out and the fielder falls down in dead ball territory, all runners are awarded one base. If the fielder remains on his feet in dead ball territory after the catch, the ball is alive and he may make a play.
Rules: 2.00 CATCH, 6.05(a), 7.04(c) Official Baseball Rules