"It's just a matter of sticking with it," said Craig Anderson, who made 33 saves for his 20th win. "We know we didn't bring our best the first two periods. We knew that if we just put together our hard hats and did the little things that we've been talking about, doing those correctly, and worrying about what things we can control going forward, we find ways to make plays."
Ottawa is on a 5-0-1 run, the only loss being a 3-2 overtime defeat at Philadelphia on Saturday.
Karlsson, who assisted on Neil's goal, finished with a plus-5 rating.
Matt Read scored a power-play goal in the second and assisted on one of three Flyers goals in the first.
Read's goal 14:08 into the second -- the third of the game by a Philadelphia rookie -- was his 13th of the season. Read gave the Flyers a 4-2 lead and tied Edmonton's Ryan Nugent-Hopkins for the most goals by an NHL rookie this season.
Bryzgalov stopped 32 shots for Philadelphia, which had won two in a row and three of four.
"I thought we turned the puck over way too many times throughout the game and I just felt that we were too soft," said Flyers center Danny Briere, who scored all three Philadelphia goals in Saturday's win. "I hate saying that, and usually it doesn't happen with the Flyers, but physically we got outplayed. Ottawa was running around way too much for my liking. That can't be happening."
Greening got credit for his ninth goal 3:11 into the third to bring the Senators within 4-3. Philadelphia defenseman Matt Carle redirected Greening's centering pass beyond Bryzgalov for the goal.
Karlsson tied it at 4 with an unassisted effort. The high-scoring defenseman took the puck away from Jaromir Jagr and drove in to beat Bryzgalov for his sixth goal.
"I don't really know what happened there," Jagr said. "I think I got the puck in my feet and then he poke-checked, and all of a sudden they went 3-on-1. I don't know where our defense went. I don't know what happened."
Jagr returned to the Flyers lineup after missing two games because of a groin injury sustained in the Winter Classic. He was limited to 7:09 of ice time in Philadelphia's 3-2 loss to the New York Rangers on Monday.
"I wanted to play because a game is different than practice," Jagr said.
Alfredsson scored the first of five goals in the first period 54 seconds in. Kyle Turris fed a pass to the slot to the Ottawa captain, who netted his fourth goal in six games.
Meszaros, who began his career with Ottawa, one-timed Max Talbot's pass and drove a shot from the right point through traffic past Anderson.
A scramble in the Flyers' goal mouth resulted in the Senators taking a 2-1 lead at 8:25. After initially signaling no goal, referee Brad Watson pointed to center ice to the delight of the sold-out crowd after a long video review determined that the puck crossed the goal line on Neil's shot into Bryzgalov's glove.
J.P. Arencibia, Brett Lawrie, Ricky Romero and Casey Janssen of the Toronto Blue Jays were in attendance. ... Ottawa will begin a three-game trip at Pittsburgh on Tuesday. ... The Senators, who host the NHL All-Star game later this month, will play nine of their next 10 games on the road. ... It was Ottawa's 14th sellout in 23 home games. ... The Flyers are 5-2 in the first game of back-to-backs this season and 3-2-1 in the second. ... The Senators are 6-14-1 when trailing after two periods.
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OTTAWA, CANADA - JANUARY 8: Ilya Bryzgalov #30 of the Philadelphia Flyers looks on during warmups prior to a game...