Forsberg tied it 1-1 just 56 seconds into the second period off of a rebound of an Alexandre Carrier shot.
"I thought we were a little flat in the first. But I thought we came out better in the second," Predators captain Roman Josi said. "Third period was kind of back and forth, some chances on both sides. We've just got to be better at the end. We gave up a goal that late, it's tough."
Hintz gave Dallas a 2-1 lead at 2:15 when he tipped in a Jason Robertson shot on the power play from the slot.
"I loved our first. I thought we played great coming out of the break, traveling this morning," Stars coach Peter DeBoer said. "I thought we had a great first. I thought in the second we gave the game back a little bit, more with puck decisions than anything. I thought we recovered and reset and got our game back in the third the way we want to play."
Trenin scored short-handed at 8:15 to tie it 2-2.
"I think we've been talking a lot about the offense and then just playing fast. I feel like the last couple of games we've done that," Josi said. "We're getting in our zone more, we're creating some more chances. It's a tough loss, but I think we've got to just keep building our game and move on."