After a scoreless second period with only seven shots for each side, Montreal mounted a comeback in the third.
Briere, a 36-year-old former all-star, was back in the lineup after sitting out as a healthy scratch for two games.
The Canadiens continued to press after his goal and put in another a few minutes later. Gionta shovelled in a loose puck at 4:46 off a point shot, after Plekanec and Briere had a crack at it first.
But it wasn't enough after Ottawa's quick start.
Greening quieted the crowd with a goal only 13 seconds into the game on a cross-crease pass from Condra.
The Canadiens had a few tough bounces to start things off.
What looked like a power-play goal by Rene Bourque at 4:05 into the first period was called back after a quick whistle on a scramble in front.
Briere finally tied the score 11:32 into the first period when he managed to knock in his own rebound after a pass from Travis Moen.
The tie didn't last long.
The Senators regained the lead on a backhand shot from Mark Stone at 13:50.
Condra scored again for the Sens 14 seconds later on a pass that went off the skate of Habs defenceman Francis Bouillon.
Heading into the second period with a two-goal lead, the Senators played a more conservative style and were able to limit the chances on Anderson. Both teams had just seven shots apiece in the second period.