Hagel made it 2-0 with a short-handed goal at 12:09, scoring with a wrist shot from the high slot that beat Bobrovsky high on the stick side.
Point extended it to 3-0 at 15:07 when he stuffed in a wraparound at the left post.
Verhaeghe cut it to 3-1 at 4:17 of the second period, chipping in a pass from Matthew Tkachuk from below the right circle.
"You give up three goals in the first ... they're a really good team over there and they are going to score goals," Verhaeghe said. "We gave a valiant effort to come back and it wasn't enough.
"It's a best-of-7 series, we're just going to try and win another game, we'll have another opportunity next game and that's all that we're thinking about."
Hagel pushed it to 4-1 at 9:40. He took a centering pass from Mikhail Sergachev on the rush and beat Bobrovsky with a shot from the high slot. Sergachev returned to the lineup after missing the past 34 games with a broken left leg.
"The body is an amazing thing because the body heals … it's that mental hurdle you have to get over," Tampa Bay coach Jon Cooper said. "And at some point, he was going to have to play a game. You can tell when a player is ready and when a player is not ready. Yesterday, we knew he was ready."
Sergachev had two shots on goal, a hit and two blocked shots in 17:03 of ice time.
"I felt great, the fans were very loud for me and I appreciate that," Sergachev said. "I felt like a hockey player again. Honestly, I couldn't really sleep last night, it felt like my first NHL game again. And then you get that from the fans in the warmup, it made me very emotional. I'm just thankful to be here."
Reinhart cut it to 4-2 at 11:10 when he went to the backhand to chip in a loose puck at the top of the crease.
Ekman-Larsson made it 4-3 at 14:33 when he beat Vasilevskiy with a wrist shot from the left circle.
"They played really well in the first period, they scored three goals in the first period," Panthers captain Aleksander Barkov said. "It's tough to come back from a three-goal lead, but we almost did. That's good for us, but obviously we don't want that to ever happen again."