"Obviously you go into every game wanting to stop every puck," Elliott said. "I was able to make all the saves I should and maybe a couple I shouldn't."
Elliott, making his 13th straight start in the absence of the injured Pascal Leclaire, has three shutouts in 41 career NHL games.
The Senators (17-12-4) also are missing top forward Jason Spezza and winger Nick Foligno. Spezza could miss six to eight weeks because of a torn medial collateral ligament, while Foligno is out indefinitely with a lower-body injury.
The absences forced Senators coach Cory Clouston to fill some holes.
One of the more creative moves was having defenseman Chris Campoli, who had been a healthy scratch the last two games, play on the wing.
"It was a great team effort," Clouston said. "Everybody did their role, everybody did their jobs and it was a nice win."
The Sabres came into the game on a four-game winning streak, but seemed to be caught off guard by the Senators, who compensated for their injury-depleted lineup with plenty of hard work.
"Not real impressive on our part," Sabres goalie Ryan Miller said. "Ottawa played a pretty solid game, but we've got to be better. ... I don't think we generated enough on the front end. They put a game plan together and did it a little bit better than us."
Picard opened the scoring in the first period, and Fisher, with his 15th, secured the win, scoring with just over a minute remaining.
"It doesn't matter how we win as long as we're finding ways and playing good defensively," Fisher said. "A bunch of guys played really well. We just need to stick to our good team game."
The Senators were also without RW Shean Donovan (knee, eight weeks). The Sabres were missing C Nathan Gerbe (groin, day-to-day), C Paul Gaustad (knee, one week) and D Chris Butler (ankle, indefinitely).