Tampa Bay moved into first place in the Atlantic Division, with one game in hand over the Buffalo Sabres, who also have 96 points.
"We can't sustain this forever, this is four games where we've been down early, we've got to play a full 60," Hagel said. "Love the comeback, love the compete, but at the same time we've got to pull a full 60 together."
Dylan Cozens and Jordan Spence scored for the Senators (38-25-10), who had points in five straight (4-0-1). James Reimer made 23 saves.
Ottawa is one point behind the Columbus Blue Jackets for the second wild card into the Stanley Cup Playoffs from the Eastern Conference.
"We pride ourselves on playing 60 minutes," Senators coach Travis Green said. "I didn't like giving up that third goal. I thought we didn't manage the puck that well in the offensive zone a few times and it stung us."
Cozens gave Ottawa a 1-0 lead 18 seconds into the game with a shot from the low slot that beat Vasilevskiy five-hole.