Tyler Pitlick scored, and Carter Hart made 17 saves for Philadelphia (10-6-3), which lost in regulation for the first time in eight games (5-1-2).
"[The Senators] kept skating. They kind of spread the ice, and they were attacking with a lot of speed," Flyers forward Jakub Voracek said. "We were a little bit slow, and we came up short."
Pitlick scored his first goal with the Flyers to give them a 1-0 lead at 2:46 of the first period. He beat Nilsson with a high wrist shot from the right face-off dot.
"We came ready to play. Our first period was good," Philadelphia coach Alain Vigneault said. "Our inability to score the second goal, when we had a couple of good looks, and their ability in the second period to win some 1-on-1 battles and to make some plays. They made a lot more plays than we did in that second period."
Ennis beat Hart five-hole with a snap shot on the power play to tie it 1-1 at 13:12 of the second, and Chlapik scored with a snap shot from the slot to give Ottawa a 2-1 lead at 17:09.
Senators forward Colin White was given a double minor for high-sticking Claude Giroux with 4:17 remaining in the third. Nilsson made three saves during Philadelphia's four-minute power play and another on Oskar Lindblom with 13 seconds left.
"In the end, we had a lot of good looks, we just didn't connect," Voracek said. "Obviously, if you don't score, it doesn't matter how good you move the puck, and if you move the puck badly and you don't score, you don't score. Especially in a 2-1 game with a four-minute power play, you should be able to find a way."