It was the teen's second goal in a week after his effort in the Carling Cup.
Rice believes the youngster can follow in the footsteps of England winger Theo Walcott, out injured tonight, to play a big role for both club and country in the years ahead.
"From Arsenal supporters' point of view, they are going to be seeing a lot of this boy," Rice said.
"He has a big, big challenge to now get in front of Theo. I know that Theo is a very strong-willed guy as well and he won't give in easy. It all bodes well for England anyway."
Rice added: "Alex can go inside, he can go outside, he's got that injection of pace and I think what he needs now is to be consistent in his play.
"I am sure that is something he will be working on because he's certainly not a stupid boy.
"He has good people around him and they are telling him all the right things.
"Alex is very friendly with Theo and no doubt he will give him the benefit of his experience as well."