Armstrong was named man-of-the-match on the night.
The 17 year-old later told the club's website: “It's always good for confidence; it's another win, and it's another step in the right direction, so hopefully we can continue in that way.
"(Peterborough) played out really well and gave us a different challenge to what we're used to, but I'm glad to get through to the next round. It was a very disciplined performance.
“I think that's the main point – we’re through to the next round. Everyone did their job and It's always an honour to wear the Everton shirt and even be involved in the squad. Another great experience against a good side and an experienced side."
Armstrong also said: “You obviously have the FA Youth Cup that you're involved in when you're Under-18s levels, so you sort of get a bit of a taste for it and to be able to even experience this competition is unbelievable."
“I'm learning something different every day,” he added. “The lads make it really easy for me. They test me at times, but in total, something new every day.”