Carlson has played all of his nine NHL seasons for Washington.
"I think it makes it that much more sweet, knowing what we've been through to get here and the path that led us here," the 28-year-old said. "It's a lot of emotions, a lot of work. A lot of time spent trying to achieve this goal. To be able to finally do it is great."
Capitals coach Barry Trotz said he thought Carlson took his game to the next level when defenseman Matt Niskanen missed 13 games after injuring his left hand in the fifth game of the season.
"When [Niskanen] went out, early, all of a sudden Johnny had to play 30 minutes a night," Trotz said. "He did a great job being able to manage it, being in all those high-pressure situations when you're exhausted. The top guys have to do that. They have to manage their game when maybe they don't have the juice in their legs or had some long shifts.
"All that being said, he's just grown as a young man. Just as [Alex Ovechkin] has grown and [Nicklas Backstrom] and all the guys … what I love about the whole group."
Asked what he and his teammates planned on doing to celebrate, Carlson said, "We'll see … the world is our oyster."