Lindgren finished a cross-ice pass from Adam Fox at the edge of the crease to tie it 2-2 at 15:52 on a 4-on-4.
"It just goes to show that when you're down 2-0 like that, late in a game, you never know what could happen," Lindgren said. "We get a power play and that gets us on the board, and we just never stopped fighting. I think that's such a big component of our team; we've got 20 guys out there that compete and never believe that we're out of games."
Goodrow tipped a shot from Sammy Blais for a 3-2 lead at 17:57.
"After the second period, we knew we weren't playing the best," Goodrow said. "We knew that we were going to have to string together a good 20 (minutes) to even have a chance to tie it up or come out with the win, so I felt like we really put our work boots on in the third period."
Paul took a centering pass from Tim Stutzle and scored with a snap shot from the slot to give the Senators a 1-0 lead 41 seconds into the first period.
"That's a game we should've had," Paul said. "Not the greatest third and we let the momentum swing."
Norris scored on a 4-on-4 with a wrist shot from the top of the right face-off circle, which went under Georgiev's right pad, to make it 2-0 at 1:04 of the third.
"I thought we carried the play for 54 minutes," Smith said. "Made some bad plays and ultimately end up losing a game that you shouldn't lose."
NOTES:The Rangers won each game on a road trip of at least four games for the first time since Feb. 10-16, 2015. … New York has its most points (eight) through its first five road games since 2010-11, when it also had eight points in five games. … Senators forward Shane Pinto did not play and will be out at least a week because of an upper-body injury. He was replaced by Parker Kelly, who had no shots on goal and three hits in 9:02 of ice time.