Bo Horvat had a goal and an assist for the Canucks (15-16-2), who have won three straight games and all five against the Senators this season.
Ryan Dzingel and Josh Norris scored for the Senators (10-20-3), who are 1-3-2 in their past six games.
"It says a lot," Ottawa defenseman Thomas Chabot said of the Senators tying the score after trailing 2-0 in each of the past two games. "Like I said the other day, we show a lot of character and we don't give up on any games and we play until the end."
Joey Daccord, making his fourth straight start for Ottawa with Matt Murray injured, allowed two goals on 21 shots before leaving at 5:07 of the third period with an undisclosed injury. Gustavsson made eight saves in his NHL debut.
"He came in and gave us a chance," Senators coach D.J. Smith said of Gustavsson. "That's all you can ask for from a goalie, to give you a chance to win. I thought Joey [Daccord] did that, then [Gustavsson] came in cold, and he had not been playing. And to make the saves he was making in the shootout, he was good."
Vancouver swept the two-game set, winning 3-2 in overtime here Monday.
"This is obviously a good stretch, not only for myself, but for the team," Demko said. "I knew that I had some good games in me, and I knew that obviously we didn't have a great start to the year, so we had to start getting wins and getting them quick."
Boeser took a pass from Horvat on a 2-on-1 and scored with a wrist shot to the blocker side to give the Canucks a 1-0 lead at 2:29 of the first. Horvat redirected a pass from Nils Hoglander on a 3-on-1 to make it 2-0 at 8:26.
Dzingel scored on the rebound of Clark Bishop's shot to cut it to 2-1 at 3:00 of the second. Norris' snap shot off the post from the slot tied it 2-2 at 5:34 of the third.
Forward Jimmy Vesey, claimed by Vancouver off waivers from the Toronto Maple Leafs on Wednesday, took the warmup but was a late scratch.
"We wanted him to play," Canucks coach Travis Green said. "A lot of hands on deck, trying to get him in. (Senior director of hockey operations) Jonathan Wall was working all day to try to get his work permit changed over, but they couldn't get it done in time."
NOTES: Ottawa forward Colin White left the game in the third period after falling into the boards. Smith did not have an update on White or Daccord and said the Senators would know more in the morning. … Chabot played 34:50, the second-highest total of his NHL career. It was the third time in four days that he played more than 29 minutes. … Canucks forward Tyler Motte played 20:42 and had two shots, four hits and four blocked shots in his first game since Feb. 6, when he sustained an undisclosed injury against the Maple Leafs. He missed 17 games.
