Manchester City attacker Jack Grealish and Arsenal's Declan Rice struck for the 2-0 victory.
Carsley said: "I thought it was good in parts. You've seen some of the things we've tried to do.
"The lads have to take a lot of credit, all of the credit. I think we've showed we've got some real talent and we can be really pleased with the clean sheet as well.
"I thought they (Rice and Grealish) took their goals really well. Really well-worked moves and brilliant finishes. I think, moving forward, they can both add a lot more goals to their game.
"Jack definitely doesn't have anything to prove to us, we can see his quality, but today will have done him the world of good, getting another goal for England and hopefully he can continue that form.
"On a really dry sticky pitch, it's not ideal for dribblers but I thought Anthony (Gordon) was a real threat. He's carried on his form from last season and he looks a really dangerous player.
"I think you've got to give Ireland credit as well, they are very resilient, they closed the gaps and our space maybe became a bit wider (in the second half).
"We found it a little bit more difficult to get through them but at 2-0 we were quite comfortable."