Hagel put Chicago ahead 1-0 7:15 into the second period, scoring from the right side of the net after a Seth Jones shot caromed off the end boards.
Hagel left the game at 9:46 of the third period with an apparent upper-body injury after he lost his balance and fell awkwardly in the Predators zone.
Carrier scored unassisted at 9:05 of the second to tie it 1-1. The rookie defenseman skated along the goal line and backhanded his first goal of the season.
Saros was tested after Predators defenseman Roman Josi was penalized for tripping at 5:55 of the third period, making saves against Kane and DeBrincat early in the Blackhawks power play and later on close-range shots by Kirby Dach and Dominik Kubalik.
"It comes to structure and details," Saros said. "I think that we have shown that even if there's been chaotic situations this year in games we've been able to stick together. I think we've handled those very well so far."
Kane became the fourth United States-born player to score 1,100 NHL points (409 goals, 691 assists), joining Mike Modano (1,374), Phil Housley (1,232) and Jeremy Roenick (1,216), and the third Blackhawks player, joining Stan Mikita (1,467) and Bobby Hull (1,153).
"As players, you always feel like you could have done more to make that situation not happen," Kane said of Colliton's firing. "But came in today and I thought (assistant) Marc Crawford and 'Kinger' did a good job of getting the message across that we wanted to play free, we wanted to play on our instincts and also play with a lot of energy, which I thought we did tonight. But it's definitely always I guess sad when something like that happens."
Chicago forward MacKenzie Entwistle left the game with 9:10 remaining with an undisclosed injury.