Villa were beaten in a two-legged semi-final by then League Two side Bradford last season but goals from Andreas Weimann, Christian Benteke and Fabian Delph ensured there would be no repeat at Villa Park on Wednesday.
"You have got to earn the right to win the football match but I thought we would with the team we put out," Lambert said. "Some of the football was excellent. The third goal - the one-touch football - was brilliant.
"There is a different feeling this time to Bradford. The squad is really together and strong and we trusted them to go out there and perform and they have been excellent.
"Cup football is a difficult thing because you can play against a small team and they are a totally different animal to what you are used to playing in the league. But we handled it well."
The victory sets up a third-round home tie with Tottenham and Lambert is happy to be back at Villa Park.
He added: "Any cup game you play you want to be at home, but we are up against a top, top side. It will be a tough game, but that's what you have to do.
"It will be great, the stadium will be full. But it's not like we have had a tough game for a couple of weeks is it?"
Although Rotherham were well beaten in the end, they competed well in the first half and manager Steve Evans felt his side had to score first if they were going to cause a shock.
"To create cup upsets you have to go in front, it's Roy of Rovers to think you can come from 2-0 down," he said. "We had to go in front and put pressure on Villa so it was very hard went we went behind.
"For 25 minutes we were by far the better side, we had two or three good chances, dominated possession, but we made a mistake and they had the quality to hurt us.
"It's two mistakes for the first two goals, we have to accept that the first one was an incredible finish and the second one, well there aren't many better players than Benteke to head that in at the far post.
"When they got the second it was a long night for us and we were chasing the ball a lot."