"We've used Dan a couple of times before and he's always had some pretty good moves," said Rangers coach Alain Vigneault. "Shootouts — who knows, eh? You throw guys over the boards and it worked out for us tonight."
Eight of Ottawa's 17 games have needed overtime or a shootout. The Senators are now 3-3 in shootouts and 1-1 in the 3-on-3 overtime.
"We hung with them for 65 minutes and when it comes down to the skills competition it's tough to let a point get away," said Senators forward Bobby Ryan.
Chris Kreider scored the lone goal in regulation for the Rangers, who have won eight in a row and have a point in 12 straight games.
Henrik Lundqvist stopped 22 shots for his tenth win of the season.
"It's one of those things both teams weren't afraid of losing and both teams wanted to win," Karlsson said. "There were quality scoring chances and two good goaltenders and a couple of posts and we go to a shootout."
The Rangers outshot the Senators 10-7 in the second period and nearly took the lead in its final seconds, but Anderson stopped Mats Zuccarello to preserve the 1-1 tie.
Kreider took exception to Zack Smith's hit on Derek Stepan late in the period. Stepan was forced to go to the quiet room as he was hit in the head, but did return and said he was fine after the game. Stepan is hoping the league will review the hit.
"I'm just happy I'm healthy," he said. "I never saw him coming."
The teams exchanged power-play goals in the first period. Kreider opened the scoring at 2:31 as he tipped Kevin Hayes' shot past Anderson.
Ottawa tied the game four minutes later when Karlsson finished off a nice passing series to beat Lundqvist.
"I thought it was a real good game, fast game," said Senators coach Dave Cameron. "We had a pretty good idea how they were going to play and it was our challenge to match and I felt we did a pretty good job."
Karlsson now has three goals in his last three games after going the first 14 without a goal.
Notes: The Senators held a moment of silence and played France's national anthem before the start of the game to honor those killed in Friday's terrorist attacks in Paris. ... D Jared Cowen missed the game because of a bruised foot and LW Shane Prince was a healthy scratch for Ottawa. ... D Dylan McIlrath and RW Emerson Etem were each a healthy scratch for the Rangers.