Niederreiter gave Carolina a 2-1 at 11:09 with a power-play goal, scoring on a rebound.
Aho scored shorthanded at 18:08 for a 3-1 lead. He finished a 2-on-0 with Martin Necas after a turnover at the Carolina blue line.
"I thought I had a breakaway there, but [Necas] is fast," Aho said. "He caught me and it was a 2-on-0. It was kind of surprising. Obviously, you don't get many of those. It was kind of a funny moment, but happy to be able to score on that."
Barkov made it 3-2 at 12:02 of the third period with a wrist shot from the slot on a 5-on-3 power play.
Florida nearly tied the game with 3:18 remaining when Jonathan Huberdeau shot on a rebound, but Nedeljkovic made a glove save.
"The penalty kill came up huge and that was the key for the game, and for sure [Nedeljkovic]," Necas said. "He played an unbelievable game."
Aho scored a shorthanded empty-net goal at 19:44 for the 4-2 final.
"Every win is huge, especially against teams like Tampa [Bay] and Florida," Necas said. "These games feel kind of like playoff games. It was a huge game. You've got to build on that and maybe next game don't take as many penalties. It's pretty hard to defend those."
NOTES: Florida defenseman Keith Yandle played his 914th consecutive game, tying Garry Unger for the second-longest streak in NHL history. Doug Jarvis has the longest streak at 964 games. … Brind'Amour did not have an update on Martinook after the game. … Carolina defensemen Brett Pesce played 28:18 and Jaccob Slavin played 27:05 . … Necas had three assists. … Barkov is the first Panthers player to have six 20-goal seasons. Olli Jokinen and Nathan Horton each had five.