Noel Hunt gave Reading a fifth-minute lead, but United came in to the game in the following half-hour only to see Mikele Leigertwood give the home side a two-goal cushion with a fine 25-yard drive against his former club.
Hunt's second, and a second-half strike from Garath McCleary, put the result beyond doubt in the second half.
"If you look at the results today, a couple of big teams have gone out and we are thankful we are not one of them," the Reading manager said.
"I don't think we did particularly well in the first half. We just weren't ourselves. I got a bit agitated, which is unusual for me.
"But we were much more clinical and efficient after, so we deserved to go through."