Oliver Bjorkstrand scored for the Lightning (18-13-3), who have lost three straight (0-2-1). Andrei Vasilevskiy, who returned after missing the previous seven games with an undisclosed injury, made 18 saves.
"Our execution [at home] just hasn't been the way we execute on the road," Tampa Bay coach Jon Cooper said. "We outshot and attempted them like 78-33. I don't know how much more we could do, but some of it comes down to execution. When we had a chance to execute we really didn't, and when we did execute the goalie made the big save."
Bjorkstrand gave the Lightning a 1-0 lead on a power play at 12:54 of the first period after Anthony Cirelli found him open in the low slot. Forsberg stopped the first shot, but Bjorkstrand put the rebound through his five-hole.
"Obviously, disappointed in the result. We haven't done a good enough job this year of making this a tough place to play," Tampa Bay defenseman Ryan McDonagh said. "For whatever reason we just didn't quite have that killer instinct, that urgency to keep putting the gas pedal down on them."
Kempe tied it 1-1 at 5:21 of the second period when he skated into the offensive zone and beat Vasilevskiy with a long-distance wrist shot from between the circles.
"I had lots of room in the neutral zone. I just tried to skate it in the O-zone and then their 'D' kind of just backed off, so I just tried to find the spot between his legs to shoot in," Kempe said. "It just happens off instinct. You don't try to go in and shoot through guys all the time. I know it's hard for the goalie when you do that. That was something I was aiming for."