Olli Jokinen also scored for the Panthers, who improved to 7-1-2 in their last 10 games. Ed Belfour stopped 31 shots.
Nolan Pratt and Vaclav Prospal scored for Tampa Bay, which lost for the second time in its last 11 games.
Tampa Bay goalie Johan Holmqvist was replaced by Marc Denis just 5:25 into the game after the Panthers scored on their first two shots.
Horton used defenceman Dan Boyle as a screen to score his 23rd goal at 5:03 on a wrist shot while skating down the right side. Only 22 seconds later, Jokinen scored his 30th on a backhander from the right circle.
Pratt made the score 2-1 at 8:16 of the second period, directing in a cross-ice pass from Martin St. Louis for his first goal.
Weiss scored twice in the second period. He split between Boyle and Paul Ranger, skated in on Denis and put a wrist shot stick side for his 15th goal at 11:11. He also scored on a slap shot from the top of the left circle at 15:38.
Peltonen was awarded a penalty shot at 7:54 of the third period after being accosted by Boyle, while skating in on Denis. Peltonen scored his 12th goal on a wrist shot that beat Denis glove side.
Peltonen became the first Florida player to score on a penalty shot at home. Horton was the last Panther to convert a penalty shot when he scored at Philadelphia on Jan. 8, 2004.
Jokinen has scored in four straight games and has at least 30 goals in three of the past four seasons.
Jacques Martin became the 12th NHL with 1,000 games. Martin is (470-386-143) in 13 years with St. Louis, Ottawa and Florida.
Notes: The Lightning have lost six of their last seven games at Florida. ... Florida's Martin Gelinas and Alexei Semenov both had two assists. ... Holmqvist had won 13 of previous 17 starts.