Schmid made 25 saves in the final two periods.
"He played a heck of a game. I mean, he had a heck of a series," New Jersey coach Lindy Ruff said. "I mean, you envision things, but you don't really envision what he did. This was a tough series. It was like I said this morning, it was two good teams going at it, and goaltending on both sides turned out to be a big factor."
Michael McLeod, Tomas Tatar, Erik Haula and Jesper Bratt scored for the Devils, who are the No. 2 seed from the Metropolitan Division and advanced for the first time since 2012. Ondrej Palat and John Marino each had two assists.
New Jersey will play the Carolina Hurricanes, the No. 1 seed from the Metropolitan, in the second round.
"It's a huge series, a rival team," McLeod said. "It just shows us how good a team we actually are and how prepared we were to come in here. And we got better every game, I think, and especially the first two (losses), I think really cleaned up our penalty kill and stuff, so it's exciting going forward."
RELATED: [Complete Devils vs. Rangers series coverage]
Igor Shesterkin made 20 saves for the Rangers, who were the No. 3 seed from the Metropolitan.
"Every year you have that same feeling, you don't know how many kicks at the can you're going to get," New York captain Jacob Trouba said. "Tomorrow is never given, especially in this league. It definitely stinks. It feels like we let an opportunity slip away.
"That's hockey. Give them credit. It's not just this room failing, I think that team played well. They had an incredible regular season. They're a good hockey team. I give them a lot of credit as well."
The Rangers had a 2-0 lead in the series, winning Games 1 and 2 in New Jersey by identical 5-1 scores.
"Maybe we took our foot off the gas when we got a 2-0 lead and gave them a little momentum," New York defenseman Adam Fox said. "But they're a fast team. They come in waves. Got to give some credit to them. It obviously feels like a missed opportunity for us.
"When we played well, you saw the score was indicative of it, but it seemed like when we couldn't score we got a little stagnated and weren't able to get one by."