Kakko gave the Rangers a 1-0 lead at 8:47 of the first period with his third goal of the season. New York outshot Ottawa 10-0 before Mikkel Boedker's chance at 11:23 of the first period.
But the Senators rallied following a fight between Mark Borowiecki and Brendan Lemieux with 6:17 remaining in the first. They scored the next five goals in a 20-minute span that started at 14:56 when Pageau tied it 1-1, scoring from the slot after Anthony Duclair missed on a wraparound.
"Once we got back, I thought we started putting pucks behind them, started shooting more pucks, started blocking shots, started hitting them," Pageau said. "I think a couple battles, a couple fights that gave us the momentum. Guys stepped up for themselves and for their teammates."
Ennis made it 2-1 with a 5-on-3 power-play goal at 18:31. Tkachuk helped create the 5-on-3 opportunity by baiting Ryan Strome into a roughing minor at 17:07 and then moving his feet to force Brady Skjei to trip him 49 seconds later.
The Rangers seemed frustrated by Tkachuk all game, and it culminated with Lemieux going after him and drawing a 10-minute misconduct with 6:56 remaining in the third period.
"He's an agitating player and we played right into his hands, that's all you can say," Quinn said. "On top of being an agitating player, he's a really good player. This game is about managing emotions and not getting distracted, and we had a few too many guys get distracted."
Hainsey scored from the slot off a backhand pass from Tkachuk one second after another power play expired to make it 3-1 at 4:12 of the second period.
Pageau gave the Senators a 4-1 lead at 11:43 on a breakaway he created by blocking Pavel Buchnevich's shot with his shins, chipping the puck up the ice and outracing everybody to it inside the far blue line.
Tkachuk scored from the slot to extend the lead to 5-1 at 14:56.
Panarin's power-play goal made it 5-2 of 17:34 of the second period.
Namestnikov, traded by the Rangers to Ottawa on Oct. 7, scored a power-play goal at 5:35 of the third period for the 6-2 final.
"Once the rough stuff started, that kind of dragged us into the battle and we played hard," Smith said.