Jacob Trouba scored, and Henrik Lundqvist made 35 saves for the Rangers (9-9-2), who are 2-3-0 in their past five games.
"An abysmal game in so many ways," New York coach David Quinn said. "Give Ottawa a lot of credit; I thought they played hard, I thought they played smart, they won every foot race, they won every 1-on-1 battle. At the end of the day, that's been the trend for us: When we start feeling good about ourselves, we don't do a really good job of handling it. This is as disappointing as it's been because I thought we were actually going in the right direction and building off a couple of real good games for us, but it's back to square one."
Anders Nilsson made 30 saves to improve to 6-1-0 in his past seven starts.
"Our goalies are doing a [heck] of a job for us, and they're finding all ways to make that big save to keep us in the game," Chabot said of Nilsson and Craig Anderson.
Chabot scored with a high wrist shot to the short side from the left face-off circle to give the Senators a 1-0 lead 57 seconds into the game.
Brown, playing his 15th game for Ottawa, ninth this season, scored from the slot on the power play off a pass from Jean-Gabriel Pageau to make it 2-0 at 6:32 of the first period.
Trouba scored with a point shot that went in off Nilsson's stick to cut it to 2-1 at 17:20.
Tyler Ennis stick-handled around two Rangers on the rush and scored with a snap shot over Lundqvist's glove to give the Senators a 3-1 lead at 1:11 of the second.
Duclair one-timed a pass from Pageau on a 5-on-3 power play to make it 4-1 at 13:24.
New York, which was coming off a 4-1 win against the Washington Capitals on Wednesday, went 0-for-4 on the power play. Ottawa was 2-for-5.
"I feel like I'm a broken record at this point because I talk about it after every win, and again after every loss following a pretty good win," Rangers forward Chris Kreider said. "We know the formula, and we buy in one game and get away from it the next. It's been like this the last few weeks. We've got to find a way to stop the bleeding."
