Bjorkstrand extended it to 5-1 with a wrist shot at the left post at 6:19 of the second period.
"When you score one, you want to get another but really it starts with the defensive part and shutting them down," Bjorkstrand said. "We're able to execute, go north, and kind of build on our chances that way."
Jesper Bratt scored a power-play goal on a breakaway to pull the Devils to within 5-2 at 13:12.
"They got some pucks that went in early, and it started being uphill right from the get-go, pretty much," Bratt said. "Obviously, not the start we wanted."
Hagel scored with a snap shot during a delayed penalty to push the lead to 6-2 at 16:35.
"When we're playing our best hockey, we're not spending too much time in our defense zone, we're playing simple down there," Lightning forward Brayden Point said. "When the plays are there to be made, we're making them but if they're not, we're flipping them out."
Angus Crookshank scored his first goal for the Devils to make it 6-3 at 3:13 of the third period, but Point made it 7-3 at 7:55.
"Obviously, pucks are going in right now, which is great to see," Point said. "We've made some really solid plays, and while not necessarily the prettiest, just good, hard plays and have been able to spend some time in the other teams' defensive zone. If you spend enough time there and guys are willing to shoot the puck, it will go in."
Cotter scored to make it 7-4 at 15:28, but Bjorkstrand scored again just 29 seconds later, this time a power-play goal at 15:57 for the 8-4 final.
"Some nights you can't buy one, and then some nights they just go in for whatever reason," Tampa Bay coach Jon Cooper said. "I think the players obviously like when they're going in, but giving up four was probably a little too many if we want to keep having success."
NOTES: Tampa Bay has scored six goals in consecutive road games for the fourth time in franchise history. The Lightning won 6-1 at the Montreal Canadiens on Tuesday. ... Tampa Bay defenseman Victor Hedman, who was placed on injured reserve Thursday, will get a second opinion on the undisclosed injury he sustained in Tuesday's win, which was his third game back after missing 12 with an undisclosed injury. ... Bratt scored his first goal in 17 games.