Geekie scored on a cross-slot pass from Zacha in front to give Boston a 3-2 lead at 1:06 of the second period.
Norris tipped a Jake Sanderson point shot on the power play to tie it 3-3 at 5:41.
John Beecher's point shot went under the right arm of Merilainen to make it 4-3 at 14:38.
"That's what's special about this year for our group: Every game matters," Swayman said. "That's something we can build on and rally around. We're putting ourselves in great opportunities to really catapult the position we're in."
Vinni Lettieri extended the lead to 5-3 at 11:23 of the third period.
"I've never played a playoff game before, but I feel like that emulated it pretty solid tonight," Ottawa captain Brady Tkachuk said. "There's been some pretty awesome atmospheres, but I don't know, this one just felt different. It felt like a preview of what's to come."
NOTES: The Senators won after overcoming a two-goal deficit in the final five minutes of regulation for the third time in their history (Feb. 13, 2023, vs. Calgary Flames; Nov. 5, 1993, at Winnipeg Jets). … It was Pastrnak's third straight game with three points. He is the first Bruins player to do that since Ray Bourque in 1993-94. … Bruins forward Brad Marchand did not take a shift in overtime; Sacco said it was a coach's decision. … Senators goalie Linus Ullmark has not played since Dec. 23 because of a back injury, but is day to day and will travel with the team on their upcoming road trip, which begins at the New Jersey Devils on Sunday.