Axelsson scored on Boston's fifth shootout attempt after Tim Thomas stopped all five Ottawa shooters in the Bruins' 4-3 victory Thursday night.
"They were better than us tonight," Axelsson said. "We were lucky to get away with two points."
Axelsson secured Boston's fourth straight victory when he put a backhander between Brian Elliott's pads after the Ottawa goalie stopped Boston's first four shooters.
"I wanted him to open up a little bit and, fortunately, I slid it under him," Axelsson said. "It was a little lucky but we'll take it."
Chuck Kobasew drew Boston even at 3 with a power-play goal with 4:02 left in the third. Kobasew put a rebound past Elliott from the goalmouth during a double minor to Vermette to tie it after the Bruins had allowed three goals to blow a 2-0 lead.
"It was a relief, to be honest with you, because we came into the third with good intentions but our energy level wasn't there tonight," Bruins coach Claude Julien said.
"We looked tired and rightfully so. We've played a lot of big games lately and we didn't need the energy to compete against a team that had lots of it and was determined to win. The bottom line is we found a way to win."
Blake Wheeler assisted on Kobasew's goal after opening the scoring with his 15th goal on a power play 11:46 in. Boston captain Zdeno Chara, who also assisted on Kobasew's goal, made it a 2-0 lead with his 12th midway through the second.
Ottawa's Nick Foligno scored later in the second and Daniel Alfredsson had a goal and an assist in the third, setting up Jason Spezza's power-play goal at 8:33 that put the Senators up 3-2.
"As much as maybe we didn't do a good job in the third, I think they did a good job of creating scoring chances," Wheeler said. "It's uncharacteristic of us to give up a lead like that but at the same time, I think you've got to give them credit."
Thomas made 19 saves.
Elliott stopped 26 shots in his 10th start in 11 games.
Elliott made a pair of big saves on Bruins defenseman Dennis Wideman in overtime, including a big glove save 1:10 into the extra period as Boston failed to take advantage of a power-play opportunity after Ottawa's Mike Fisher drew an interference penalty at the end of regulation.
Thomas robbed Fisher with a sprawling pad save later in overtime.
Wheeler opened the scoring for Boston with his 15th goal on a power play 11:46 in.
Chara made it 2-0 midway through the second. The Boston captain drove a slap shot from the point past Elliott for his 12th goal at 10:27.
Ottawa lost its fourth game in a row overall, and second straight under interim coach Cory Clouston.
"It's obviously disappointing to lose but we did some great things tonight and I think that's the most important thing," Foligno said. "We've just got to continue to do those things and I thought we took it to them for the majority of the night."
The Senators had been shut out twice in their previous three games, including a 1-0 loss to Los Angeles on Tuesday night in Clouston's NHL debut.
Foligno scored a highlight-reel goal 15:54 into the second.
After receiving Bell's pass at the Boston blue line, Foligno drove the left side then cut toward the slot, moving past defensemen Andrew Ference and Wideman to reach around and put a backhand past Thomas before tumbling along the ice.
Alfredsson scored his 15th goal 58 seconds into the third to tie it at 2.
Bruins LW Milan Lucic didn't play after sustaining an undisclosed foot injury Wednesday night in a 3-1 win in Philadelphia. ... Wideman's points streak ended at seven games. He had four goals and four assists during that span. ... Senators C Dean McAmmond was a healthy scratch for the second time in four games.