Erik Karlsson scored in the shootout to give Ottawa a 2-1 victory Thursday, with New Jersey falling to 0-12 in the tiebreaker this season.
Karlsson snapped a shot past goalie Cory Schneider on the Senators' fourth attempt for the only goal.
"I just stuck with it and, luckily for me, it worked," Karlsson said.
New Jersey's Dainius Zubrus failed to score on Lehner after Travis Zajac, Adam Larsson and Jagr were denied. The Devils have lost 16 shootouts in a row dating to last season, the worst streak in NHL history.
"It's not like we didn't try," Jagr said. "I've got to blame myself. I had those chances to finish the game in overtime, even before the overtime; it just didn't happen."
New Jersey has not won a shootout since March 10, 2013, when Patrik Elias scored to give the Devils a 3-2 win against the Winnipeg Jets.
"Let's say something positive," Jagr said. "I believe, and it goes so bad right now, that something great is going to happen. Maybe not this year, but next year."
Michael Ryder drew New Jersey even at 1-1 with 2:34 remaining in the second period. Ryder scored his 18th goal well after a delayed penalty had been signaled against Ottawa.
Mike Hoffman gave the Senators a 1-0 lead with his third goal at 11:20.
Lehner made 38 saves in regulation and overtime, including Jagr's breakaway with 42 seconds remaining.
"I think he missed his shot a little bit," Lehner said. "He tried to go over the pad, he hit it, but he's a great player. It's fun to see him come down the ice and you want to stop him."
Schneider made 31 saves in his fifth straight start in the Devils' last road game of the season. New Jersey finished 14-18-9 on the road.
"We win half the shootouts, we're right in it," Schneider said. "We score six more goals here or there, we win games we may have lost, but we've got to do something because it seems that after 80 games it's been a pretty consistent theme of losing games that are winnable, and we've got to make some adjustments."
Devils coach Peter DeBoer said he liked most of the effort he saw from his players in the wake of failing to make the Stanley Cup Playoffs for the third time in four seasons.
"You could see the first 10 minutes we were a little bit out of it," DeBoer said. "I think that getting eliminated [Wednesday] night, I think there was a little bit of a hangover, but I thought once we started to get into the game and start to move, I thought we carried the play and we should have had more than one goal."
Ottawa forward Mika Zibanejad, who assisted on Hoffman's goal, was taken to a hospital after sustaining an undisclosed injury. Zibanejad did not play after ending his second shift of the third period at 4:51.
"I have no idea what it's all about, but it was for purely precautionary reasons that we couldn't do stuff here that they could do there," Senators coach Paul MacLean said.
Elias returned after missing the past two games. The forward left New Jersey's 2-1 win against the Washington Capitals in the first period Friday after he hit his head on a stanchion when he was checked by Tom Wilson.
Ottawa's Ales Hemsky was awarded a penalty shot by referee Dan O'Halloran at 10:48 of the second period after he was hooked on a breakaway by New Jersey defenseman Marek Zidlicky. Hemsky's shot went wide to the left.
Hoffman scored 32 seconds later to give the Senators a 1-0 lead. Zibanejad set up the goal at 11:20 when he carried the puck past the goal line and put a centering pass back to the slot, where Hoffman jumped up to put away his first in 11 games.
Senators left wing Colin Greening returned after he was sidelined for six games because of a lower-body injury.
Hemsky's penalty shot was the Senators' second of the season, and the third called against the Devils. Ottawa's Kyle Turris failed to score on a penalty shot against Steve Mason of the Philadelphia Flyers on Nov. 19.
Schneider allowed a penalty-shot goal by Derek Stepan of the New York Rangers on Jan. 26. Martin Brodeur stopped J.T. Brown of the Tampa Bay Lightning on March 15.
The Devils host the New York Islanders at Prudential Center on Friday and finish the regular season at home Sunday against the Boston Bruins.
The Senators play their final home game Saturday against the Toronto Maple Leafs. Ottawa ends the season Sunday on the road against Pittsburgh Penguins.