Patrice Bergeron and Brad Marchand also scored for Boston, which improved to 6-1-1 on the road.
After blowing a two-goal lead, the Bruins took a 3-2 advantage on Marchand's goal from the slot with 6:02 to go in the second. Horton extended the lead to 4-2 when he redirected Zdeno Chara's pass at 2:33 of the third.
Tampa Bay got goals from Steven Stamkos and Cory Conacher, but the Lightning's two-game winning streak, which followed an 0-5-1 stretch, was snapped.
The Bruins took a 2-0 lead early in the first on goals by Horton and Bergeron.
Horton scored on the rebound from the low left circle 1:11 into the game. Bergeron made it 2-0 when he beat goalie Anders Lindback from the right circle at 3:50.
Horton has 26 goals and 46 points in 45 games against the Lightning.
Tampa Bay got within 2-1 on Stamkos' 11th goal of the season with 11:47 left in the first. Teddy Purcell set up the goal after winning a battle for the puck in the slot.
Purcell finished with two assists.
Conacher got a wrist shot from the left circle past Tuukka Rask 9 minutes into the second that tied it 2-2. The rookie snapped an eight-game drought with his sixth goal of the season.
Notes: Boston D Dougie Hamilton had two assists. ... Stamkos has four goals in three games. ... Boston LW Milan Lucic was back in the lineup after he missed Sunday's 3-2 win at Winnipeg because of a personal matter.