Flyers forward Claude Giroux was hit with a double-minor penalty for high-sticking with 6:07 left. Greening buried a wrister past Ilya Bryzgalov for his eighth goal seconds later for the 2-1 lead.
The Senators won for the first time since March 28. They had lost the first four games of a seven-game trip and claimed sixth place in the Eastern Conference. They are tied in points with the New York Islanders, two points in front of the eighth-place New York Rangers.
Daniel Alfredsson scored an empty-net goal with 6.9 seconds left. Zack Smith also scored for Ottawa.
Giroux had the lone goal for the Flyers.
Much like they have for most of the season, the Flyers struggled to generate offense. The Flyers' four-game winning streak earlier this month stirred hope they could make a late playoff push. But they scored two goals total in the last two games, both losses, and entered Thursday seven points out of a playoff spot. They have now scored three goals in three games.
With eight games left, the Flyers are running out of time.
Philadelphia forward Scott Hartnell said after Thursday's morning skate that the team would need an "epic ending" to make the playoffs. This wasn't a great start.
They took only 14 shots through the first two periods and needed Bryzgalov and a stout penalty kill to keep this one tight.
Bryzgalov had come under scrutiny this week for shutting his eyes and yawning on the bench during a loss last weekend to Winnipeg. He was later anonymously accused by some teammates of sleeping - or at least pretending to doze off - during a team meeting on Monday.
Flyers coach Peter Laviolette is on the hot seat, three years after leading the team to the Stanley Cup finals. On April 11, 2012, the Flyers beat the Penguins 4-3 in Game 1 of an Eastern Conference quarterfinal series. The Flyers won that series in six games.
Giroux was touted by some as the "best in the world" after a sensational postseason and was named captain entering this season. His production has dipped as have the numbers of just about every key Flyers player.
Scott Hartnell and Danny Briere have missed chunks of time because of injuries, and they could have a long offseason to recover. With a hefty salary and declining production, Briere knows the Flyers could use one of their amnesty buyouts on him.
Bryzgalov and Briere's futures will be decided in the offseason. For the Flyers, that is coming sooner than they would like.
Smith was alone on a short breakaway attempt, and backhanded the puck past Bryzgalov for his fourth goal.
After an interference penalty was whistled against Ottawa, the Flyers capitalized on the short-handed attempt. Giroux tipped in Simon Gagne's shot to make it 1-1.