New York contained Alex Ovechkin again and completed its comeback after trailing in the series 2-0 and 3-2 — the latest in Washington's long history of playoff collapses.
Sixth-seeded New York faces No. 4 Boston in the second round.
It is the first time New York won a Game 7 on the road in its history.
Arron Asham put New York ahead in the first period, before Taylor Pyatt and Michael Del Zotto made it 3-0 early in the second on goals 2:10 apart.
Ryan Callahan added a goal 13 seconds into the third period, and when Mats Zuccarello scored with about 13 1/2 minutes remaining, thousands of red-clad fans streamed to the exits.