Conklin, acquired from Columbus on Feb. 27, stood tall for the Sabres who were outshot 45-20.
Chris Drury broke a 3-3 tie at 9:50 of the third period when he tipped in Brian Campbell's slap shot from the point.
Drew Stafford scored twice for Buffalo, including an insurance goal at 15:17. Jason Pominville and Dainius Zubrus also had goals for Buffalo, which hadn't won since beating Toronto on March 3. Campbell had three assists.
Nathan Horton had two goals for Florida, which lost at home in regulation for only the second time in 11 games (8-2-1). Stephen Weiss added a goal and two assists for the Panthers. His linemate Martin Gelinas had two assists.
Ed Belfour, who tied a franchise record with his 22nd consecutive start, stopped 15 shots.
Buffalo, which leads the Eastern Conference with 96 points, rallied from deficits of 1-0 and 2-1 and took a 3-2 lead into the third period.
Horton tied it at 2:45 when a wrist shot from the point was deflected to the side of the net. He then one-timed it past Conklin.
Conklin kept the Sabres even with a couple of big saves off deflections, including a tip by Florida centre Olli Jokinen.
After Drury put Buffalo in front, Stafford made it 5-3 when he one-timed Campbell's pass from behind the net.
Weiss' goal gave Florida a 1-0 lead after the first period, during which the Panthers held a 15-5 shots advantage.
The Sabres answered with three goals in the second period. Stafford gave Buffalo its first lead of the game when he finished off a pretty three-way passing play by tipping in Campbell's cross-ice feed.
Notes: Roberto Luongo had set Florida's record for consecutive starts last season. ... Buffalo already led the NHL with 104 second-period goals before adding three against Florida. ... It was the fourth time this season the Sabres allowed at least 40 shots. ... The line of Weiss, Horton and Gelinas has 32 points in Florida's last 11 games.