Hagel scored, and Mikhail Sergachev had two assists for the Lightning (39-21-6), who won for the second time in their past eight games. Brian Elliott made 22 saves.
Tampa Bay is third in the Atlantic Division, four points behind the second-place Toronto Maple Leafs.
"In the end it's a 1-1 game with five minutes to go and you win it in regulation," Lightning coach Jon Cooper said. "That's got to be a step forward. I liked how we pushed at the end. It would have been nice to do that the whole game. We didn't, but when it counted we did."
Taylor Raddysh scored, and Petr Mrazek made 34 saves for the Blackhawks (22-38-6), who have lost seven of eight and were coming off a 4-3 overtime loss at the Florida Panthers on Friday.
"Overall, I thought our game was pretty good," Chicago forward Tyler Johnson said. "I thought we had a lot of chances, and Mrazek was unbelievable and kept us in it. It was a close game. ... It came down to the end there. I wish that wouldn't happen, but you have to take some positives."
Point gave the Lightning a 1-0 lead at 6:21 of the second period when he stuffed in his own rebound after Victor Hedman found him open in the slot.
"Unbelievable player," Hedman said of Point. "He scores a lot of goals from the slot but in different ways. Different kinds of goals today, one was a little lower and one was a little higher. He's always in the right spot."
Raddysh tied it 1-1 at 5:23 of the third with a one-timer from the left face-off circle off Cole Guttman's pass from below the goal line.
"I think our guys are a little bit deflated after two nights of great effort against two great teams down here," Blackhawks coach Luke Richardson said. "We just have to reassure [them] that they did a lot of things right and that we're going in the right direction.
"We'll learn from this. When you turn the corner, the lessons of losing in the last minute turn into winning in the last minute."