It was the first victory for Florida over Atlanta in seven games this season.
The loss snapped a five-game win streak for the Thrashers.
Olli Jokinen and Nathan Horton also scored goals for Florida and Ed Belfour stopped 28 shots for the Panthers.
Marian Hossa and Keith Tkachuk scored for Atlanta. Kari Lehtonen had 36 saves for Atlanta.
With the teams skating 4-on-4, the Panthers stretched their lead to 3-1 on Bouwmeester's goal at 3:52 of the third period. His shot from just inside the blue line went through Lehtonen's legs.
Tkachuk closed the gap to 3-2 at 15:29 of the third. Belfour blocked a shot by Hossa but the puck stayed in the crease. Tkachuk poked it past Belfour on the glove side.
The Panthers scored two goals in just over two minutes to take a 2-1 lead in the second period. Jokinen's shot from the right circle found the net when the puck went through a narrow space between Lehtonen's glove and the post to tie the score 1-1 at 5:48.
Horton gave the Panthers a 2-1 lead after his shot from above the right circle went over Lehtonen's left leg and into the net at 7:38.
Atlanta took a 1-0 lead at 9:28 of the first. Alexei Zhitnik took a shot from the left circle. Belfour stopped it but Hossa grabbed the rebound and beat Belfour glove side. It was Hossa's team-leading 40th goal. Hossa extended his point streak (seven goals, 10 assists) to 10 games, a new Thrashers team record.
Notes: Hossa reached the 40-goal mark for the second time in his nine-year NHL career. ... Hossa has seven goals and three assists in seven games against Florida this season...Belfour started his 20th consecutive game for Florida...