Mike Fisher, Chris Kelly, Jason Spezza and Nick Foligno scored for the Senators, who matched the Washington Capitals for the NHL's longest active win streak. It is Ottawa's longest run since Oct. 13-Nov. 6, 2007.
Elliott, who began the season as the team's backup, has been in net for the last six of those victories. He has allowed only six goals during that span.
Evgeni Malkin scored 1:34 into the game for the Penguins, who had won three of four but had their lead over the Senators for fourth place in the Eastern Conference standings trimmed to three points.
The Senators, whose winning streak was preceded by five losses in a row, have outscored opponents 27-8 during the past eight.
With No. 1 goalie Pascal Leclaire injured throughout much of the season, Elliott has taken advantage of the playing time. One of his best saves Thursday was denying Sergei Gonchar from the slot during a power play early in the second period.
He shut down Pittsburgh after Malkin scored his sixth in six games early on a perfectly placed shot from the right circle that beat Elliott high to the glove side.
But the closest the Penguins would come to scoring the remainder of the game came when Kris Letang shot the puck into the net moments after the horn had sounded to end the second period.
Fisher tied it at one at 10:06 of the first with his 18th, a tap-in into an open net off a feed from Foligno moments after Fisher's wrist shot had hit the post.
The Senators took the lead for good 3:33 later when Kelly tapped a loose puck into an open net while standing in front of the crease, following a shot by Jarkko Ruutu for his 10th of the season.
Spezza scored for the third consecutive game since returning from missing 20 games because of a knee injury, his eighth of the season, 4:27 into the third.
Foligno added an empty-netter.
Notes
The game was Penguins C Jordan Staal's 300th. … Pittsburgh LW Ruslan Fedotenko had the 300th point of his career. … Foligno has a three-game point streak. … Ottawa is 12-1-1 when leading after one period and 19-1 when leading at the second intermission.