The Reds boss singled out the younger members of his squad after two goals from Nuri Sahin saw them through to the fourth round.
"It was an outstanding performance," he said. "I'm not surprised as our performance level has been good all season. It's great to see the kids playing with confidence.
"We fully deserved the win. For me it's about talent irrespective of age. We've got a small squad.
"It's a club where young players are going to get a chance. It's congratulations to Jerome Sinclair. Our future is bright with the young players. The style and definition of our game is improving all the time."
Liverpool will play Rodgers's former club Swansea City at Anfield in round four.