Elliott, named the NHL's First Star of the Week on Monday, made 24 saves to win his fifth straight start. It was his fourth career shutout.
"It's not so much against him, it's against the team and they play a patient game and pounce on opportunities when you give them to them," Elliott said. "And we really didn't give them opportunities tonight and I think that was the key to the victory."
Kovalev and Michalek scored in the first to stake Ottawa to a 2-0 lead. Spezza scored the Senators' third goal 15:47 into the second. Filip Kuba had two assists.
New Jersey, which has been held to 11 goals while losing five of seven overall, had won nine straight against Ottawa.
"We're not skating the way we can skate and when that happens we're not as good with the puck, we're not as confident, you know, we don't create as well," Devils coach Jacques Lemaire said. "You look at our chances -- we didn't get a lot of chances and most of them, we made it easy for the goalie. We made it very easy."
Brodeur, who stopped nine of 12 shots in two periods, saw his personal winning streak against the Senators end at six games, including the first three meetings between the teams this season.
"I think we let them dictate the game," Brodeur said. "I think they definitely are playing well but for us it's not acceptable the way we competed against a team like them. But they played well, they did the right things defensively and offensively to win this hockey game."
Yann Danis replaced Brodeur to begin the third. He stopped all six shots he faced.
"We lost three games against them earlier this season so it was nice to get three goals and it was nice to see him get out," Michalek said. "They probably wanted to rest him but it was nice to see."
New Jersey's skid has dropped the Atlantic Division leaders to second place overall in the Eastern Conference with 69 points, three behind Washington and two points ahead of both Pittsburgh and Buffalo.
Chris Campoli, who played his 300th game, recorded an assist on Michalek's goal -- originally credited to Spezza -- 18:42 into the first.
Brodeur, the NHL's career leader in wins and shutouts, robbed Michalek with a sensational glove save on Ottawa's first shot before Kovalev opened the scoring midway through the first.
The Russian right wing took Nick Foligno's pass and put a shot through Brodeur's pads for his 13th goal at 9:12.
The Senators had an apparent second goal disallowed at 17:16. Ottawa's Chris Phillips put a shot off the left post and Jonathan Cheechoo kicked the rebound back towards the net. The puck struck Brodeur's blocker and was redirected into the net just inside the left post.
Referee Marc Joannette waved off the goal following a video review. NHL Rule 49.1 (i) states: "A kicked puck that deflects off the body of any player of either team [including the goalkeeper] shall be ruled no goal."
Ottawa got its 2-0 lead 1:26 later when Michalek deflected Campoli's wrist shot from the point on its way past Brodeur.
Spezza scored for the second game in a row since returning from a knee injury that sidelined him for 20 games in Saturday's 2-1 win in Boston.
Brodeur, who hadn't lost in Ottawa since Oct. 8, 2007, is 34-19-4 overall against the Senators. ... Danis made his seventh appearance of the season, his fourth in relief of Brodeur. ... Campoli, who played for the New York Islanders from 2005-09, has 28 goals and 82 assists for 110 points overall in his career. ... Ottawa set a team record with eight straight wins from Oct. 13-Nov. 6, 2007. ... Senators backup G Mike Brodeur, a distant relative of Martin Brodeur, won the first two games of Ottawa's current streak.
Copyright by STATS LLC and The Associated Press
OTTAWA - JANUARY 26: Chris Neil #25 of the Ottawa Senators stands outside the locker room before taking warm-ups...