Carroll struck in the 3-1 win.
Rodgers said of Carroll on Liverpool's official website: "It was a terrific goal from him. It was a nice, clever piece of play by Charlie Adam as the ball came into him and he flicked it around the corner.
"Maybe eight or nine times out of 10 he would have expected that (the ball) back, but big Andy showed his skill and ability as well.
"He's got stick that he can't play this way, but it was a wonderful piece of skill letting the ball go through his legs, getting the shot on target and tucking the ball away very well. It was very pleasing all round.
"The players had a wonderful idea of the game tactically and I am seeing an improvement every single day in terms of our passing game."