Lundqvist had been in goal for the Rangers while they were losing three consecutive first-round playoff series over the last four years, but he came through against the Senators in Game 7.
The Rangers will take on Washington in the Eastern Conference semifinals after the Capitals eliminated defending champion Boston in their Game 7 showdown Wednesday.
Marc Staal and Dan Girardi both scored their first goal of the playoffs during a span of 4:18 of the second period to give the Rangers a 2-0 lead.
Daniel Alfredsson brought the Senators within one by putting a slap shot past Lundqvist with 8:26 to go in the middle period and that finished the scoring.
New York, which earned the No. 1 seed in the Eastern Conference, is seeking its first Stanley Cup title in 18 years and only its second since 1940.
Craig Anderson allowed the two New York goals on 29 shots.
The Rangers faced elimination in two straight contests, surviving the sixth game with a 3-2 win in Ottawa. Five of the seven games in the series were decided by one goal and the other two by two goals.
New York's victory kept the NHL from having both its No. 1 seeds eliminated in the first round. Vancouver, No. 1 in the West, was knocked out by Los Angeles.