Sauer scored his first goal on a power play with 5:19 left to break a tie, and New York went on to beat the Senators 5-3 on Thursday.
"It's something you're always going to remember," Sauer said. "At every level you remember your first one and it was great that it was a game-winning goal, too. I'm definitely excited."
Carkner got a misconduct added to his roughing major after a fight with Rangers enforcer Derek Boogaard at 17:29 of the first.
Rangers coach John Tortorella refused to comment on what the Ottawa defenseman did as he passed by New York's bench after the fight. Some Rangers players suggested that Carkner sent blood toward them.
"I heard something like that, that he was swiping blood and flicking it at our guys," forward Brandon Dubinsky said. "I'm not sure 100 percent."
Carkner denied the accusation.
"I was getting off the ice, and maybe I made a gesture," he said.
Erik Christensen drew the Rangers even for the third time earlier in the third period. Dubinsky scored into an empty net with 51.9 seconds remaining.
Henrik Lundqvist stopped 24 shots for the Rangers, who have won seven of 10 despite a 3-1 loss to the Senators on Sunday in the first half of the home-and-home series.
Ottawa has been outscored 39-19 in the third period.
"We need to clean up that area for sure," said Senators forward Nick Foligno, who had two assists. "Third periods are when games are won, so we've got to maybe be a little bit more resilient, you know, dig a little bit deeper. That comes from within the team."
Christensen netted his fifth goal 3:30 into the third when his shot from the left corner somehow got past Elliott, who was hugging the right post.
"It was a really good effort coming back three times," Lundqvist said. "I think we were just determined to win this one after a tough loss at home where they got the late goal there. It felt really good the way we responded."
Lundqvist made a sparkling glove save off Foligno on Ottawa's first shot of the game, after Michal Rozsival was penalized for high-sticking Senators captain Daniel Alfredsson 1:33 in, but Foligno passed the rebound to Fisher, who fired in his eighth goal.
Stepan tied it 1:46 later with his seventh.
Spezza made it 2-1 with Ottawa's second power-play goal 34 seconds into the second period. Fedotenko tied it at 2 with his fifth goal 69 seconds later.
Fisher appeared to make it three goals in 1:34 when he sent a shot past Lundqvist at 2:08, but referee Paul Devorski's call was overturned when a replay showed that Fisher's shot struck the crossbar and bounced onto the ice without crossing the goal line before caroming off the left post.
Fisher's second of the game put Ottawa up 3-2 at 8:47. It was his ninth this season.
- Rangers D Michael Del Zotto was scratched for the first time this season. Del Zotto has two goals and six assists in 29 games.
- The Senators had scored fewer than two goals in eight of 12 games.