Artem Zub and Tim Stutzle scored for the Senators. Forsberg had 13 saves in the first period, 12 in the second and 10 in the third to get his first win since Dec. 14 after going 0-2-1.
"We're shutting down their best plays every shift," Forberg said. "I didn't really think they were able to create a lot. I think it was a really good job, and overall everybody fought hard."
Sean Kuraly scored for Columbus, and Elvis Merzilikins had 31 saves. The Blue Jackets lost for the third time in four games overall, and third straight at home.
Zub put Ottawa on the board with a wrister from just inside the blueline at 7:59 of the first.
"The first period it looked like they were really skating and we weren't, but we hung on and got some timely goals and we found a way to check ourselves to a big win," Ottawa coach D.J. Smith said.
Kuraly tied it with 34 seconds left in the opening period, deflecting Jakub Voracek's shot from the right circle.
Ottawa then pulled ahead when Stutzle outskated Zach Werenski and slipped a backhander over Merzlikins with 3:58 left in the second.
"It was a fairly even game," Blue Jackets coach Brad Larsen said. "It's not like we were chasing the game. It was a little bit lifeless. I was hoping to have more energy from our guys. We didn't have the extra gas. We needed more energy and we didn't have that."
HELLO, OLD FRIEND
The game was the first meeting between Columbus and Ottawa in nearly two years. The teams last met Feb. 24, 2020, when visiting Columbus won 4-3 in overtime.
Columbus activated defenseman Vladislav Gavrikov and center Cole Sillinger off the COVID-19 protocols list, and assigned defenseman Jake Christiansen to the taxi squad. Adam Boqvist was cleared off COVID-19 protocols and practiced Sunday and is expected to return Wednesday against Calgary.
FORMENTON ON FIRE
With the primary assist on Stutzle's goal, Alex Formenton recorded his seventh point in his last six games. He now has 10 assists and 17 points.
Sunday night's game finishes off the Senators' seventh back-to-back series this season, as they improved to 5-7-2.
Senators: Host Buffalo on Tuesday night.
Blue Jackets: Host Calgary on Wednesday night.