The deal was announced two days before Wednesday's NHL trade deadline and gives the Senators a skilled pest with plenty of playoff experience.
Burrows, after waiving his no-trade clause, agreed to a two-year, $5 million extension with the Senators. He was set to become an unrestricted free agent this summer. A source told ESPN's Pierre LeBrun that Burrows also receives a 10-team modified no-trade clause in his new deal.
Burrows, who turns 36 in April, played 822 regular-season games with the Canucks, with 193 goals, 191 assists and 1,066 penalty minutes.
The undrafted free agent who fought his way from the third-tier ECHL to eventually star on Vancouver's top line added 19 goals and 15 assists in 70 playoff games.
"We had to do something like this," Senators general manager Pierre Dorion said Monday before Ottawa's game in Tampa, Florida. "We just felt it was time to add another piece, and in Alex Burrows, I can tell you, we're ecstatic to have added that piece."
Burrows had four straight seasons of at least 25 goals from 2008-09 to 2011-12 playing largely with Henrik and Daniel Sedin. His best season came in 2009-10 when he scored 35 goals and added 32 assists.
"It's been a hard day," Canucks general manager Jim Benning said in Vancouver. "Alex Burrows has been the heart and soul of our franchise for a long time.
"He's given us everything that he's had. He's been so good with our young players this year, but given where we're at right now we just felt it was time to make the move.
"We got back, we feel, a good, young player in Jonathan Dahlen, so we're excited about that."
Burrows' days as an offensive catalyst are behind him -- he has nine goals and 11 assists in 55 games this season -- but he will still bring an edge to a Senators team pressuring the Montreal Canadiens for the top spot in the Atlantic Division.
Dahlen, 19, was selected in the second round, 42nd overall, by Ottawa at the 2016 draft.
The forward has impressed at the international level, notching six points in seven games for Sweden at the 2017 world junior championship and leading his country in scoring with five points in five games at the 2016 Ivan Hlinka under-17 tournament.
"He's a player we've had our eye on," said Benning, whose team didn't have a second-round pick last year. "He's a goal scorer, he's good around the net, he's sure-handed, he's got a good nose for the net and he'll pay the price to get there.
"Where we're at right now and where Ottawa's at, it was a fair deal for both teams."
Information from The Associated Press was used in this report.