Bednar scored 2:52 into the third period and Luongo made 29 saves as the Panthers rallied for a 2-2 tie with the Toronto Maple Leafs.
After Leafs defenseman Aki Berg swatted down a centering pass by Viktor Kozlov, Bednar intercepted Berg's clearing attempt in the slot. The rookie fired it back over the right pad of goaltender Trevor Kidd to knot it.
It was the second comeback in as many games for Florida, which rallied from two third-period deficits in Friday's 3-3 tie with the New York Islanders.
"We need to get some winning games," Bednar said. "It's just about confidence. We've just got to work hard. We just need one (big) game, then everything is going to be all right."
Toronto, which has not won on the road away since November 29, is winless in its last four games (0-3-1-0) and 3-7-2 on the road.
"We had good chances to win and we just aren't mentally strong enough," Maple Leafs coach Pat Quinn said. "I said this the other night, we're not mentally tough enough right now to take a game that maybe you should have had. We should have had this game tonight. We should have had the one in Atlanta. Shouldas don't count though."
Florida also had a chance to end its four-game homestand on a positive with six minutes to go in the third period. Rookie Stephen Weiss stole the puck from McCabe behind the net and centered it to Niklas Hagman, who was alone in front but put a shot off the goalpost.
"I think he lost the puck a bit and rolled off his stick right onto mine," Weiss said. "It was bouncing like crazy, so I just chipped it over to (Hagman) and I don't know if Kidder stopped it or what, but it was a good chance."
Luongo kept Florida within a goal by fending off a point-blank opportunity by Darcy Tucker. H also denied Alexander Moginy on a reakaway in the second peri and Nik Antropov on two close-ranchances erly in the third.
Kidd, o matched Luongo with acrobatic sav stopped 0 shots.
"He played a woderful gae and we just don't seem toe able to score some goals for him to support the performance that he's throwing out," Quinn said. "He's been very good and he's certainly not the reason we didn't take two points out of here tonight."
"They were probably both equal in their games tonight, and that's what I think you should ask of a goaltender," Panthers coach Mike Keenan said. "Be as good as the other guy at the other end and your team has to win some games for you."
Held to two power-play goals in their previous seven games, the Leafs scored on their first two chances in this one.
Mats Sundin gave Toronto the early lead midway through the opening period. Panthers defenseman Sandis Ozolinsh tipped a cross-ice pass by Mogliny to Sundin at the left side of the crease, where he flipped the puck over a diving Luongo for his 16th goal.
Mogilny set up the Leafs' second goal with 2:09 left in the second period as Tie Domi tipped in a pass for his eighth of the season.
But rookie defenseman Jay Bouwmeester halved the Panthers' deficit 20 seconds later. The third overall pick in the 2002 draft motored across the blue line and fired a slap shot from the top of the right faceoff circle that squeezed under Kidd's arm and trickled across the goal line.