The Magpies were 2-0 up in the first leg and won the second leg by the same margin.
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They went through 4-0 on aggregate and ensured they are in a second final in several years.
Newcastle defender Dan Burn told Sky Sports: "It was a good reaction from the weekend and just buzzing to be back at Wembley. You have to trust the manager and what he tells you to do."
Team-mate Kieran Trippier added: "Arsenal do game management a lot and we did the same. If we stick to the game plan we will go far as a team. It means everything to reach the final. We have had the experience before.
"The first time there was a lot of emotion and now we know how to handle that. The ticket stuff was carnage and it takes a lot of the energy out of you."